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**Tim Maudlin**: *on* 4/24/15 at 19:37pm UTC, wrote Peter, The link you post is dead. In any case, I was making a couple of...

**Peter Jackson**: *on* 4/24/15 at 18:26pm UTC, wrote Tim, I understand your views, all conventional. You seem to take agreement...

**Laurence Hitterdale**: *on* 4/23/15 at 1:10am UTC, wrote Dear Dr. Maudlin, Thank you. Your response is helpful. I tend to favor...

**Janko Kokosar**: *on* 4/22/15 at 20:01pm UTC, wrote Dear Tim I never learn topology, so I do not understand everyting. First I...

**Pentcho Valev**: *on* 4/22/15 at 17:00pm UTC, wrote "the discussion with Valev was not productive, and irrelevant to my paper...

**Tim Maudlin**: *on* 4/22/15 at 16:31pm UTC, wrote Dear Peter, As I'm sure you know, a complete discussion of the presently...

**Pentcho Valev**: *on* 4/22/15 at 16:07pm UTC, wrote Tim Maudlin wrote on Apr. 17, 2015 @ 18:08 GMT: "I stopped the discussion...

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December 11, 2017

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Trick or Truth Essay Contest (2015)
[back]

TOPIC: How Mathematics Meets the World by Tim Maudlin [refresh]

TOPIC: How Mathematics Meets the World by Tim Maudlin [refresh]

The most obvious explanation for the power of mathematics as the language of physics is that the physical world has the right sort of structure to be represented mathematically. But what this in turn means depends on the mathematical language being used. I first briefly review some of the physical characteristics required in order to unambiguously describe a physical situation using integers, and then take up the much more difficult question of what characteristics are required to describe a situation using geometrical concepts. In the case of geometry, and particularly for the most basic form of geometry— topology—this is not clear. I discuss a new mathematical language for describing geometrical structure called the Theory of Linear Structures. This mathematical language is founded on a different primitive concept than standard topology, on the line rather than the open set. I explain how some other geometrical concepts can be defined in terms of lines, and how in a Relativistic setting time can be understood as the feature of physical reality that generates all geometrical facts. Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space: all of the geometry flows from temporal structure. The Theory of Linear Structures also provides a mathematical language in which the fact that time is a fundamentally directed structure can be easily represented.

Tim Maudlin is Professor of Philosophy at NYU. He received his B.A. in physics and philosophy from Yale and his Ph.D. in History and Philosophy of Science from the University of Pittsburgh. His books include Quantum Non-Localtiy and Relativity (Blackwell), The Metaphysics Within Physics (Oxford), Philosophy of Physics: Space and Time (Princeton), and New Foundations for Physical Geometry: The Theory of Linear Structures (Oxford). He has been a Guggenheim Fellow.

Dear Tim Maudlin,

You emphasize that associating a mathematical structure with physical items is*not* the same as postulating that they *are* mathematical entities. I fully agree with this.

As you note, boundaries and structural integrity through time is sufficient for enumeration, and, per Kronecker, given integers, the rest of math follows.

You then ask, what...

view entire post

You emphasize that associating a mathematical structure with physical items is

As you note, boundaries and structural integrity through time is sufficient for enumeration, and, per Kronecker, given integers, the rest of math follows.

You then ask, what...

view entire post

report post as inappropriate

Dear Dr. Klingman,

Thanks for your remarks. I will be reading other essays once the semester here settles down a bit and look forward to yours.

Regards.

Tim Maudlin

Thanks for your remarks. I will be reading other essays once the semester here settles down a bit and look forward to yours.

Regards.

Tim Maudlin

Tim,

Thank you for contributing an interesting essay. Perhaps I am misinterpreting your ideas, but for clarification, are you suggesting in an effort to understand the fundamentals of mathematics in physics, we use a different language or include more "descriptive fundamental concepts or words"? Furthermore, is it your argument that the physics we describe in our universe is somewhat...

view entire post

Thank you for contributing an interesting essay. Perhaps I am misinterpreting your ideas, but for clarification, are you suggesting in an effort to understand the fundamentals of mathematics in physics, we use a different language or include more "descriptive fundamental concepts or words"? Furthermore, is it your argument that the physics we describe in our universe is somewhat...

view entire post

report post as inappropriate

Dear Demond,

The issue is not so much words (although of course we have to use words to convey what we mean) but rather what the fundamental mathematical concepts are in a particular mathematical theory. The application of those concepts typically requires that certain axioms be satisfied, so the clearest way to understand why the mathematical concepts would apply usefully to the physical...

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The issue is not so much words (although of course we have to use words to convey what we mean) but rather what the fundamental mathematical concepts are in a particular mathematical theory. The application of those concepts typically requires that certain axioms be satisfied, so the clearest way to understand why the mathematical concepts would apply usefully to the physical...

view entire post

Hi Tim,

Thanks for the enjoyable essay. Although i share your view that "The physics ... physical world has the right sort of structure to be represented mathematically", but I think it is not end of the story, It is to say, in same physical world we have e.g. rise of self-consciousness in particles, that can't be represented mathematically. The other words, there are structures that can't fit into math either partially or entirely.

The essential fact is that whatever thing that has a quantity can fit into math, for example the true mechanism behind the quantum entanglement can't be observed and hence doesn't have a quantity and can't be fitted into.math. These are what I had to say in my article.

Kind regards

Koorosh

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Thanks for the enjoyable essay. Although i share your view that "The physics ... physical world has the right sort of structure to be represented mathematically", but I think it is not end of the story, It is to say, in same physical world we have e.g. rise of self-consciousness in particles, that can't be represented mathematically. The other words, there are structures that can't fit into math either partially or entirely.

The essential fact is that whatever thing that has a quantity can fit into math, for example the true mechanism behind the quantum entanglement can't be observed and hence doesn't have a quantity and can't be fitted into.math. These are what I had to say in my article.

Kind regards

Koorosh

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Dear Koorosh,

I agree that consciousness raises a very difficult problem, which I do not pretend to be able to solve. Since physics has gotten along so well dealing with non-conscious systems, there should be a way to understand that which is independent of that issue. That is all I was trying to discuss.

Regards,

Tim

I agree that consciousness raises a very difficult problem, which I do not pretend to be able to solve. Since physics has gotten along so well dealing with non-conscious systems, there should be a way to understand that which is independent of that issue. That is all I was trying to discuss.

Regards,

Tim

Dear Professor Maudlin,

You make excellent point in your essay. I want to comment on three ideas. First, on “The

Unreasonable Relevance of Some Branches of Mathematics to Other Branches”, I would recommend Connes’ essay “A view of mathematics: ”http://www.alainconnes.org/docs/maths.pdf (see page 3): “there is just “one” mathematical world, whose exploration is the...

view entire post

You make excellent point in your essay. I want to comment on three ideas. First, on “The

Unreasonable Relevance of Some Branches of Mathematics to Other Branches”, I would recommend Connes’ essay “A view of mathematics: ”http://www.alainconnes.org/docs/maths.pdf (see page 3): “there is just “one” mathematical world, whose exploration is the...

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Dear Prof. Moldoveneanu,

Thanks for the message. The comment by Connes is on target, of course. I recently was discussing my work with Shahn Majid, who also works on non-commutative geometry, and he did find commonalities and points of contact between the two. But I doubt that the two approaches will yield exactly the same formalism or results, so there is more to explore about the...

view entire post

Thanks for the message. The comment by Connes is on target, of course. I recently was discussing my work with Shahn Majid, who also works on non-commutative geometry, and he did find commonalities and points of contact between the two. But I doubt that the two approaches will yield exactly the same formalism or results, so there is more to explore about the...

view entire post

I was wondering in relationship to your question: "If it is correct, then we might see how the time itself creates the geometry of space-time" if you are aware of 1963 Zeeman's result: causality implies the Lorentz group: https://download.wpsoftware.net/causality-lorentz-group-zeem

an.pdf

Time demands causality and Zeeman's result (which was independently discovered by 2 more people) is...

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an.pdf

Time demands causality and Zeeman's result (which was independently discovered by 2 more people) is...

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Dear Florin,

Yes, the Zeeman result is familiar to me, although it is actually much too restrictive for my purposes, as it presupposes a Minkowski metric, and we want to have a theory that can describe all of the the solutions to the General Relativistic Field Equations, and even other possible geometries (e.g. discrete geometries) as well. What is easy to show in my setting is that the...

view entire post

Yes, the Zeeman result is familiar to me, although it is actually much too restrictive for my purposes, as it presupposes a Minkowski metric, and we want to have a theory that can describe all of the the solutions to the General Relativistic Field Equations, and even other possible geometries (e.g. discrete geometries) as well. What is easy to show in my setting is that the...

view entire post

Tim,

I thought your essay was interesting, though I have somewhat different ideas about things. I think spacetime is emergent from quantum entanglement. The emergence of time occurs in the Wheeler De-Witt equation HΨ[g] = 0. The wave functional is defined on an entire spatial manifold, but in general spatial slices only have diffeomorphisms with each other that define time on a local chart or patch. We may then consider the projection onto Hilbert subspaces H_i ⊂ H, P_i so that

P_i|Ψ[g]> = e^{θ_i}|ψ_i>,

which may be accomplished with a sum over other states

P_i = sum_{j=!i}|ψ_j>

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I thought your essay was interesting, though I have somewhat different ideas about things. I think spacetime is emergent from quantum entanglement. The emergence of time occurs in the Wheeler De-Witt equation HΨ[g] = 0. The wave functional is defined on an entire spatial manifold, but in general spatial slices only have diffeomorphisms with each other that define time on a local chart or patch. We may then consider the projection onto Hilbert subspaces H_i ⊂ H, P_i so that

P_i|Ψ[g]> = e^{θ_i}|ψ_i>,

which may be accomplished with a sum over other states

P_i = sum_{j=!i}|ψ_j>

report post as inappropriate

I forgot that this editor does not like carrots, so I use parentheses instead.

P_i = sum_{j=!i}|ψ_j)(ψ_j|. It is simplest of course to consider i = 1 or 2 for two different regions. The relative phase θ_i = ω_it, and defines a local time.

A simple case of this is the de Sitter spacetime with two patches in static coordinates or standard dS slicing, such as seen in the diagram below. In this case there are two patches with different time “arrows,” or better put two sets of diffeomorphisms that correspond to local spacetime.

Cheers LC

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P_i = sum_{j=!i}|ψ_j)(ψ_j|. It is simplest of course to consider i = 1 or 2 for two different regions. The relative phase θ_i = ω_it, and defines a local time.

A simple case of this is the de Sitter spacetime with two patches in static coordinates or standard dS slicing, such as seen in the diagram below. In this case there are two patches with different time “arrows,” or better put two sets of diffeomorphisms that correspond to local spacetime.

Cheers LC

report post as inappropriate

Dear Lawrence,

Your proposal is a bit compressed here, but one obvious question, apart from others, is where the ω you mention comes from. Without it, you do not define a t. In the usual case, ω is E, the eigenvalue in an energy eigenstate, and the energy eigenstates are the eigenstates of H. But of course, in Wheeler-deWitt H annihilates the state, so the eigenvalue is zero. That is the classic problem of time in Wheeler-deWitt.

Also, if you are somehow dealing with deSitter, the you already have a space-time structure (since the Ψ[g] is defined over the entire space of metrics, how did we get to de Sitter?), so the "emergence" problem must already have been solved to make sense of the rest of the construction. How, in that case, does time "emerge"?

Regards,

Tim

Your proposal is a bit compressed here, but one obvious question, apart from others, is where the ω you mention comes from. Without it, you do not define a t. In the usual case, ω is E, the eigenvalue in an energy eigenstate, and the energy eigenstates are the eigenstates of H. But of course, in Wheeler-deWitt H annihilates the state, so the eigenvalue is zero. That is the classic problem of time in Wheeler-deWitt.

Also, if you are somehow dealing with deSitter, the you already have a space-time structure (since the Ψ[g] is defined over the entire space of metrics, how did we get to de Sitter?), so the "emergence" problem must already have been solved to make sense of the rest of the construction. How, in that case, does time "emerge"?

Regards,

Tim

The WDW equation does not have any time, for spacetimes in general relativity most often do not have a single set of diffeomorphisms that include the entire manifold. The conjugate meaning of this is that with E conjugate to t there is no manner in most manifolds by which one can form a Gaussian surface to define mass-energy. Mass-energy is not localizable. This only happen for stationary spacetimes with some asymptotic flatness.

What I mentioned above is a situation where the Hilbert space for the WDWE is partitioned into two parts so that locally in the manifold corresponding to either of them time can be maybe defined. The states for the two Hilbert space subsets are entangled and this entanglement is exchanged with the occurrence of horizons separating the two regions. The horizons, such as in a black hole, are entangled with states in the exterior. This is in effect a sort of coarse graining, and spacetime is a coarse graining of entangled states. This coarse graining reflects a lack of information about the nature of the entanglement.

Cheers LC

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What I mentioned above is a situation where the Hilbert space for the WDWE is partitioned into two parts so that locally in the manifold corresponding to either of them time can be maybe defined. The states for the two Hilbert space subsets are entangled and this entanglement is exchanged with the occurrence of horizons separating the two regions. The horizons, such as in a black hole, are entangled with states in the exterior. This is in effect a sort of coarse graining, and spacetime is a coarse graining of entangled states. This coarse graining reflects a lack of information about the nature of the entanglement.

Cheers LC

report post as inappropriate

Tim,

This is a very interesting essay. I have one minor criticism. The first four pages are very philosophical. That's ok, no problem. You lay out some history, some limitations, etc ... Basically counting and geometry. Then at the end of page four you introduce the true objective of your essay. A lazy reader would be denied this new knowledge:-)

Regarding your new topology, is it necessary to use two of the same thing to create the linear structure? By this I mean can you only form a linear structure using geometric points? The reason for my question is that it appears to me that you are applying topology with its vast legacy knowledge to Hamilton's quaternions.

All in all, very well done. Thank you for sharing these new ideas.

Best Regards and Good Luck,

Gary Simpson

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This is a very interesting essay. I have one minor criticism. The first four pages are very philosophical. That's ok, no problem. You lay out some history, some limitations, etc ... Basically counting and geometry. Then at the end of page four you introduce the true objective of your essay. A lazy reader would be denied this new knowledge:-)

Regarding your new topology, is it necessary to use two of the same thing to create the linear structure? By this I mean can you only form a linear structure using geometric points? The reason for my question is that it appears to me that you are applying topology with its vast legacy knowledge to Hamilton's quaternions.

All in all, very well done. Thank you for sharing these new ideas.

Best Regards and Good Luck,

Gary Simpson

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Hi Gary,

Since I am proposing the Linear Structure as the most fundamental geometrical structure that exists, the objects over which it is defined are automatically geometrical points, by which I mean they have no intrinsic geometrical structure. I get that for free. But of course, they could differ in all kinds of other ways. They could have different intrinsic properties (as an illustrative example, think of them as having different electrical charges), but these would not be geometrical properties. The structure of these other properties would also be unconstrained by the Linear Structure over the elements. One could, I imagine, use quaternions to represent these intrinsic properties, but then the theory is going beyond geometry.

Thanks, and best wishes,

Tim

Since I am proposing the Linear Structure as the most fundamental geometrical structure that exists, the objects over which it is defined are automatically geometrical points, by which I mean they have no intrinsic geometrical structure. I get that for free. But of course, they could differ in all kinds of other ways. They could have different intrinsic properties (as an illustrative example, think of them as having different electrical charges), but these would not be geometrical properties. The structure of these other properties would also be unconstrained by the Linear Structure over the elements. One could, I imagine, use quaternions to represent these intrinsic properties, but then the theory is going beyond geometry.

Thanks, and best wishes,

Tim

Hello. I have several remarks about your work, first about general topology, then about physics.

I also considered this problem, of how else topology might be formalized so as to be more convenient for different purposes, may it be easiness of use, restrictions or generalizations, or modified kinds of structures (usually more rigid than some topology, since topology is usually the lightest...

view entire post

I also considered this problem, of how else topology might be formalized so as to be more convenient for different purposes, may it be easiness of use, restrictions or generalizations, or modified kinds of structures (usually more rigid than some topology, since topology is usually the lightest...

view entire post

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Dear Sylvain Poirier,

Thanks for the careful reading of the paper (which is, of course, very brief) and the comments. Let me try to advance the conversation.

There are a few different threads of thought here. Let's separate them.

One is how the Theory of Linear Structures interacts with category theory. It is an interesting question—when I taught a seminar on this, there was...

view entire post

Thanks for the careful reading of the paper (which is, of course, very brief) and the comments. Let me try to advance the conversation.

There are a few different threads of thought here. Let's separate them.

One is how the Theory of Linear Structures interacts with category theory. It is an interesting question—when I taught a seminar on this, there was...

view entire post

When I mentioned to consider the sets of morphisms as primitive, I meant only : in a first draft of consideration, until fixing a definition of the structures that the morphisms will preserve. Also, while not strictly necessary, I generally admit that the objects in the category are given as sets (with structures to be introduced later), and the morphisms are maps between these...

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Well, I won't try to comment on your approach, given that there is obviously not room to explicate it properly, and in any case it is not directly relevant to my paper.

There is quite a bit I did not specify in my paper, as it is a less-than-9-page introduction to something that takes hundreds of pages to present!

But to your questions: Yes, by my definitions, y=0 is indeed a neighborhood of (0,0) in that set, and that does completely align with my intuitions. Any continuous line that arrives a (0,0) has a segment in y = 0. In that sense, which is the only obvious one, the set of points y=0 completely surrounds (0,0) in that set, which is what we want a neighborhood to do. You seem to think that it is a problem that my definitions yield different results than the standard ones. In this case, mine in the more intuitive. As for Cantor sets, neither I nor anyone else has any real intuitions about them, so it is hard to argue that any result about them is either good or bad.

I should also note that you simply repeat certain properties of the standard definition as if they are desirable but without any argument. This is particularly the case with pre-images. The most natural thing is to define properties of functions by the geometrical characteristics that they preserve under their action: that is, the geometrical characteristics of the domain that are preserved in the range. The whole idea of looking at what is preserved backwards, i.e. by the pre-images. is just strange. You are used to it because it is what has been done, but it is just peculiar.

There are lots of differences between my definitions and the standard ones. You seem to think that these differences are per se objectionable. But there is no argument to that effect.

I am also a bit mystified by your assertion that you do not need to check my book to see how I have defined things. Are you psychic?

There is quite a bit I did not specify in my paper, as it is a less-than-9-page introduction to something that takes hundreds of pages to present!

But to your questions: Yes, by my definitions, y=0 is indeed a neighborhood of (0,0) in that set, and that does completely align with my intuitions. Any continuous line that arrives a (0,0) has a segment in y = 0. In that sense, which is the only obvious one, the set of points y=0 completely surrounds (0,0) in that set, which is what we want a neighborhood to do. You seem to think that it is a problem that my definitions yield different results than the standard ones. In this case, mine in the more intuitive. As for Cantor sets, neither I nor anyone else has any real intuitions about them, so it is hard to argue that any result about them is either good or bad.

I should also note that you simply repeat certain properties of the standard definition as if they are desirable but without any argument. This is particularly the case with pre-images. The most natural thing is to define properties of functions by the geometrical characteristics that they preserve under their action: that is, the geometrical characteristics of the domain that are preserved in the range. The whole idea of looking at what is preserved backwards, i.e. by the pre-images. is just strange. You are used to it because it is what has been done, but it is just peculiar.

There are lots of differences between my definitions and the standard ones. You seem to think that these differences are per se objectionable. But there is no argument to that effect.

I am also a bit mystified by your assertion that you do not need to check my book to see how I have defined things. Are you psychic?

Hi Tim,

Clearly one of the best essays yet :) I have one issue to pick. You state that the physical world has the right structure to be describable as math as if that was a fact, but forget to question whether that is indeed so. It is arguably true that some of it is describable as math. But is all of it describable by math? What if not? That's what I've addressed in my essay.

-- Sophia

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Clearly one of the best essays yet :) I have one issue to pick. You state that the physical world has the right structure to be describable as math as if that was a fact, but forget to question whether that is indeed so. It is arguably true that some of it is describable as math. But is all of it describable by math? What if not? That's what I've addressed in my essay.

-- Sophia

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Hi Sophia,

So what we know for sure is that certain aspects of the physical world have structures that can be described to very high accuracy by mathematically formulated theories: everyone uses the anomalous magnetic moment of the electron as the example of precision, and the entire standard model is tremendously successful, as is the General Theory of Relativity. It does not follow that everything can be captured mathematically, but whatever cannot must not play a very noticeable role in producing all of the phenomena that physics has been able to predict so accurately. If some aspect of the world cannot be represented mathematically, then no mathematically formulated theory will succeed predictively for it.

In any case, I did not claim that it is all describable by math (probably no mathematical theory will account for consciousness, for example) but that the geometry of space-time is. That's a hard enough problem all by itself!

Regards,

Tiim

So what we know for sure is that certain aspects of the physical world have structures that can be described to very high accuracy by mathematically formulated theories: everyone uses the anomalous magnetic moment of the electron as the example of precision, and the entire standard model is tremendously successful, as is the General Theory of Relativity. It does not follow that everything can be captured mathematically, but whatever cannot must not play a very noticeable role in producing all of the phenomena that physics has been able to predict so accurately. If some aspect of the world cannot be represented mathematically, then no mathematically formulated theory will succeed predictively for it.

In any case, I did not claim that it is all describable by math (probably no mathematical theory will account for consciousness, for example) but that the geometry of space-time is. That's a hard enough problem all by itself!

Regards,

Tiim

Dear Tim,

I just read your interesting essay, and also your comments elsewhere on the Heraclitean and Parmenidean views of reality. Your essay touches on fundamental questions in geometry and how to correct the perceived wrong conceptual foundation. I am sure others will have other questions on your interesting contribution. For me since I discuss similar interest in my essay, I will have...

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I just read your interesting essay, and also your comments elsewhere on the Heraclitean and Parmenidean views of reality. Your essay touches on fundamental questions in geometry and how to correct the perceived wrong conceptual foundation. I am sure others will have other questions on your interesting contribution. For me since I discuss similar interest in my essay, I will have...

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Dear Akinbo,

Let me try to answer your questions. If "cutting" a line just means partitioning it into two parts, each of which are themselves lines, that is partitioning a line into two segments, then this can be done for any linear order: it is exactly what Dedekind called a "Schnitt". Now Dedekind wanted something more in order to define a continuum: for every Schnitt, there should be either one or two points that correspond to the Schnitt. Think of this as either the greatest or least element of one of the two parts of the partition. Some linear orders can be cut in this way without there being such an element. For example, the set of positive rational numbers can be partitioned in two groups: those whose squares are greater than 2 and those whose squares are less than two. That is a perfectly good Schnitt, but there is no greatest element of the one part or least element of the other. So, by Dedekind's definition, the set of rational numbers is not a continuum.

I have no problem with Dedekind's definition. It just shows that lines can be defined—and cut—even if the space is not a continuum.

As for perishing: the physical lines I have in mind are sets of events, ordered by a temporal order. The universe could have a maximal element in time—a last event. That is a claim about the overall geometry of the universe. If you mean by "perishing" that any object in the past has "perished", then lines do indeed perish: lines made of events in my past have, from my present perspective, perished. That is just the same sense in which we generally talk about things in time perishing: no longer existing.

Regards,

Tim Maudin

Let me try to answer your questions. If "cutting" a line just means partitioning it into two parts, each of which are themselves lines, that is partitioning a line into two segments, then this can be done for any linear order: it is exactly what Dedekind called a "Schnitt". Now Dedekind wanted something more in order to define a continuum: for every Schnitt, there should be either one or two points that correspond to the Schnitt. Think of this as either the greatest or least element of one of the two parts of the partition. Some linear orders can be cut in this way without there being such an element. For example, the set of positive rational numbers can be partitioned in two groups: those whose squares are greater than 2 and those whose squares are less than two. That is a perfectly good Schnitt, but there is no greatest element of the one part or least element of the other. So, by Dedekind's definition, the set of rational numbers is not a continuum.

I have no problem with Dedekind's definition. It just shows that lines can be defined—and cut—even if the space is not a continuum.

As for perishing: the physical lines I have in mind are sets of events, ordered by a temporal order. The universe could have a maximal element in time—a last event. That is a claim about the overall geometry of the universe. If you mean by "perishing" that any object in the past has "perished", then lines do indeed perish: lines made of events in my past have, from my present perspective, perished. That is just the same sense in which we generally talk about things in time perishing: no longer existing.

Regards,

Tim Maudin

Dear Tim,

Thanks for finding the time to reply. Following your response, I checked on 'Schnitt', which is German for 'cut'. So as not to confuse issues, by cutting of 'a line', I do not mean mathematical cutting of the number line in Dedekind's sense. By line, I mean extension in Euclid's sense. A point cannot be cut by definition, and unlike the number line where an irrational number can be invented as a 'trick' to provide a "gap" in order for cutting to take place, on an extended line "gap" itself will connote either an extension, distance or space and therefore consist of points, all of which similarly cannot be cut. Probably, if you later read my essay you may get my meaning.

If the physical lines you have in mind are sets of events, rather than extension that I mean, then of course events cannot be cut in two.

Then on "perishing" and the possibility of your own type of line perishing, please give a thought of the implication of this in resolving Zeno's Dichotomy paradox, even though Calculus is mathematically used to find a solution to it. However, the 'infinitesimal' of calculus or "ghost of departed quantities" as is famously called challenges aspects of physical reality. Calculus does not tell us the size of the last dx in the race. Calculus cannot also explain how to cut a line of the extended type. Thanks for the exchange.

All the best,

Akinbo

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Thanks for finding the time to reply. Following your response, I checked on 'Schnitt', which is German for 'cut'. So as not to confuse issues, by cutting of 'a line', I do not mean mathematical cutting of the number line in Dedekind's sense. By line, I mean extension in Euclid's sense. A point cannot be cut by definition, and unlike the number line where an irrational number can be invented as a 'trick' to provide a "gap" in order for cutting to take place, on an extended line "gap" itself will connote either an extension, distance or space and therefore consist of points, all of which similarly cannot be cut. Probably, if you later read my essay you may get my meaning.

If the physical lines you have in mind are sets of events, rather than extension that I mean, then of course events cannot be cut in two.

Then on "perishing" and the possibility of your own type of line perishing, please give a thought of the implication of this in resolving Zeno's Dichotomy paradox, even though Calculus is mathematically used to find a solution to it. However, the 'infinitesimal' of calculus or "ghost of departed quantities" as is famously called challenges aspects of physical reality. Calculus does not tell us the size of the last dx in the race. Calculus cannot also explain how to cut a line of the extended type. Thanks for the exchange.

All the best,

Akinbo

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Dear Sir,

We have discussed Wigner’s paper in our essay to show that the puzzle is the result of unreasonable manipulation to present an un-decidable proposition and impose the unreasonableness on mathematics. We have specifically discussed complex numbers (since he has given that example) and other examples. You are welcome to read and comment on it.

Your statement: “A physical...

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We have discussed Wigner’s paper in our essay to show that the puzzle is the result of unreasonable manipulation to present an un-decidable proposition and impose the unreasonableness on mathematics. We have specifically discussed complex numbers (since he has given that example) and other examples. You are welcome to read and comment on it.

Your statement: “A physical...

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"Our understanding of the structure of time has been revolutionized by the Theory of Relativity. Intriguingly, the change from a classical to a Relativistic account of temporal structure is of exactly the right sort to promote time into the sole creator of physical geometry."

Even if the speed of light were constant, as Einstein postulated in 1905, it can be shown that no new temporal structure arises - rather, the concept of time becomes "not even wrong". But that is an obsolete argument because, as a recent experiment showed, the speed of light is not constant:

"The work demonstrates that, after passing the light beam through a mask, photons move more slowly through space."

Pentcho Valev

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Even if the speed of light were constant, as Einstein postulated in 1905, it can be shown that no new temporal structure arises - rather, the concept of time becomes "not even wrong". But that is an obsolete argument because, as a recent experiment showed, the speed of light is not constant:

"The work demonstrates that, after passing the light beam through a mask, photons move more slowly through space."

Pentcho Valev

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The question of the "speed" of light does not even arise at this level of geometrical analysis. What gets built into the Linear Structure of a Relativistic space-time is just the conformal (light cone) structure. That structure has no classical analog at all, and so constitutes something completely new in relation to classical absolute time and absolute simultaneity.

Strange argument. What could

"The question of the "speed" of light does not even arise at this level"

mean? You don't want to think of it? The postulate of the constancy of the speed of light is essential, even if all heads are in the sand, and since it is false, the return to "classical absolute time and absolute simultaneity" is unavoidable.

Pentcho Valev

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"The question of the "speed" of light does not even arise at this level"

mean? You don't want to think of it? The postulate of the constancy of the speed of light is essential, even if all heads are in the sand, and since it is false, the return to "classical absolute time and absolute simultaneity" is unavoidable.

Pentcho Valev

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The notion of the speed of anything is a metrical notion. Topology describes geometrical features of a space that are not tied so closely the metrical features: a topological feature is invariant under transformations that change metrical relations. So no speed is definable at this level of description.

Consider just the conformal structure of a Relativistic space-time. This does not contain the sort of metrical information one would need to define a speed, but does define a notion of temporal precedence and a by that a light-cone structure. Those structures are not classical. Even more, they are inconsistent with the classical account of temporal structure.

It is not a matter of what I want to think about, but of what is formally definable at a certain level of geometrical description.

Tim Maudlin

Consider just the conformal structure of a Relativistic space-time. This does not contain the sort of metrical information one would need to define a speed, but does define a notion of temporal precedence and a by that a light-cone structure. Those structures are not classical. Even more, they are inconsistent with the classical account of temporal structure.

It is not a matter of what I want to think about, but of what is formally definable at a certain level of geometrical description.

Tim Maudlin

Topology of Linear Structures is in all you've generalised, it would be more explainable if you diverse from geometrically conceived prospects, which would bring more light to the subject.

Great job & luck!

Sincerely,

Miss. Sujatha Jagannathan

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Great job & luck!

Sincerely,

Miss. Sujatha Jagannathan

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Dear Tim,

I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation. Also as something you can confirm directly from Roger Penrose being a fellow FQXi member...

Here is what Roger Penrose has to say in his book,*The Emperor's New Mind*, p.113… "The system of real numbers has the property for example, that **between any two of them, no matter how close, there lies a third.** It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to **divide up** the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (…10^{-35}m), this would indeed be the case".

Hence, my asking assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are*uncuttable* into parts?

Regards,

Akinbo

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I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation. Also as something you can confirm directly from Roger Penrose being a fellow FQXi member...

Here is what Roger Penrose has to say in his book,

Hence, my asking assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are

Regards,

Akinbo

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Dear Akinbo,

Geometrical points have no geometrical parts, by definition. Hence, a single point cannot be further divided. That is true whether the points on a line are dense (there is always a point between any other two) or not dense. Roger Penrose (and George Ellis, for example) think that in this sense physical space or space-time is not dense. My own mathematical language can handle spaces that are dense and spaces that are not dense.

A distance can be divided, as Dedekind shows, by partitioning a line into two sets of points in certain way. This does not require dividing any individual point in two.

Regards,

Tim

Geometrical points have no geometrical parts, by definition. Hence, a single point cannot be further divided. That is true whether the points on a line are dense (there is always a point between any other two) or not dense. Roger Penrose (and George Ellis, for example) think that in this sense physical space or space-time is not dense. My own mathematical language can handle spaces that are dense and spaces that are not dense.

A distance can be divided, as Dedekind shows, by partitioning a line into two sets of points in certain way. This does not require dividing any individual point in two.

Regards,

Tim

Dear Tim,

I do like your theory of linear structures and it does look as if it could have application in four dimensional spacetime in which a series of points on a line in four dimensional spacetime can include variability in the space and time dimensions.

Can you describe any applications or experiments in which this theory has been used?

Regards

Richard

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I do like your theory of linear structures and it does look as if it could have application in four dimensional spacetime in which a series of points on a line in four dimensional spacetime can include variability in the space and time dimensions.

Can you describe any applications or experiments in which this theory has been used?

Regards

Richard

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Dear Richard,

What I have developed is not itself a physical theory but rather a mathematical language in which physical theories can be written. The language provides clues about, for example, how to describe a discrete space-time that has Relativistic characteristics. But no complete novel theories have as yet been formulated in this mathematical language, since it has just been developed. Nonetheless, one can see how this language could be well-adapted to describing the physical world on account of its temporal structure.

Regards,

Tim

What I have developed is not itself a physical theory but rather a mathematical language in which physical theories can be written. The language provides clues about, for example, how to describe a discrete space-time that has Relativistic characteristics. But no complete novel theories have as yet been formulated in this mathematical language, since it has just been developed. Nonetheless, one can see how this language could be well-adapted to describing the physical world on account of its temporal structure.

Regards,

Tim

Dear Tim,

I had a further thought about applications of the Theory of Linear Structures. Can you generalize the theory so that it can progress from dealing with sequential points on a line to points on a plane or curved surface and then on to points in three dimensions and the four dimensions of spacetime.

The idea is to try to show a mathematical (or topological?) equivalence between String theory which models oscillating strings of one dimension in a hypothetical 12 dimensional spacetime (11 space plus one time) with the Spacetime Wave theory which proposes oscillations in spacetime (wave propagation of changes in spacetime curvature at speed c) as the description of photons and fundamental particles.

This objective is referenced briefly in my essay on Solving the Mystery.

Regards

Richard

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I had a further thought about applications of the Theory of Linear Structures. Can you generalize the theory so that it can progress from dealing with sequential points on a line to points on a plane or curved surface and then on to points in three dimensions and the four dimensions of spacetime.

The idea is to try to show a mathematical (or topological?) equivalence between String theory which models oscillating strings of one dimension in a hypothetical 12 dimensional spacetime (11 space plus one time) with the Spacetime Wave theory which proposes oscillations in spacetime (wave propagation of changes in spacetime curvature at speed c) as the description of photons and fundamental particles.

This objective is referenced briefly in my essay on Solving the Mystery.

Regards

Richard

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Dear Richard,

The Theory as it is deals with as high-dimensional spaces as you like. One specifies the Linear Structure of, say, a n-dimensional space-time by specifying those sets of events that constitute continuous lines. If the theory only worked for one-dimensional spaces it would not be worth much.

Maybe you mean this: the 2-D world-sheet of an open string can be thought of as a sequence of lines. Can one extend the machinery used to describe sequences of points that constitute lines to cover sequences of lines that constitute worlds sheets? I have not thought about that. One problem, of course, is that the worlds sheet can be partitioned into sequences of lines in different ways, and none is "the right" war to do it. There is no similar ambiguity when resolving a line into a sequence of points.

Regards,

Tim

The Theory as it is deals with as high-dimensional spaces as you like. One specifies the Linear Structure of, say, a n-dimensional space-time by specifying those sets of events that constitute continuous lines. If the theory only worked for one-dimensional spaces it would not be worth much.

Maybe you mean this: the 2-D world-sheet of an open string can be thought of as a sequence of lines. Can one extend the machinery used to describe sequences of points that constitute lines to cover sequences of lines that constitute worlds sheets? I have not thought about that. One problem, of course, is that the worlds sheet can be partitioned into sequences of lines in different ways, and none is "the right" war to do it. There is no similar ambiguity when resolving a line into a sequence of points.

Regards,

Tim

" The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, "

I'm not sure about either one of these two hypotheses. It is a realist view that does not conform with the standard model of quantum mechanics. It complies with the tensless view of relativity but these are different theories and definitively not fully united other than in QED. Can we be sure that the laws of physics will not change one million years from now? Obviously, if we say they will not, this is an axiom and far from a common sense truth.

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I'm not sure about either one of these two hypotheses. It is a realist view that does not conform with the standard model of quantum mechanics. It complies with the tensless view of relativity but these are different theories and definitively not fully united other than in QED. Can we be sure that the laws of physics will not change one million years from now? Obviously, if we say they will not, this is an axiom and far from a common sense truth.

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I can't imagine what you have in mid by this comment. If you think you can change fundamental physics by your command, you are welcome to try. And if you are unable to change the mathematical language you use to write physical theories, then you are oddly constrained. I did not say that the laws won't change (I do not mention laws anywhere here), but in any case they will not change at our command. so if we want our mathematical theories to describe the sort of structure the world has, we have to change the theories to fit the world rather than the other way around.

"If you think you can change fundamental physics by your command, you are welcome to try"

What I tried to say and maybe I did not say it correctly is that your comment " The physical world is as it is, and will not change at our command." involves a hypothesis that there is something we can call fundamental physics. You or anyone else have not proved that. I can assume that we live in a virtual reality in which the laws were changed by its creators. There is an infinite regression of fundamental physics founded on fundamental physics. Except if you are talking about fundamental assumptions like particles. I think Einstein changed fundamental physics by his command, meaning the fundamental physics that people thought they were. Speaking about absolute fundamental physics makes no sense imo. It's like speaking about unicorns. Because it is true that: Two unicorns imply 1+1 = 2

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What I tried to say and maybe I did not say it correctly is that your comment " The physical world is as it is, and will not change at our command." involves a hypothesis that there is something we can call fundamental physics. You or anyone else have not proved that. I can assume that we live in a virtual reality in which the laws were changed by its creators. There is an infinite regression of fundamental physics founded on fundamental physics. Except if you are talking about fundamental assumptions like particles. I think Einstein changed fundamental physics by his command, meaning the fundamental physics that people thought they were. Speaking about absolute fundamental physics makes no sense imo. It's like speaking about unicorns. Because it is true that: Two unicorns imply 1+1 = 2

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Ah. By "the physical world" I mean the physical world, not our theories about the physical world (however widely accepted they may be). In this sense, Einstein did not change the physical world at all: he developed new theories about that world. Even if we live in a "virtual reality", then it is a physical fact about us that we do. Probably, we could never figure that out.

If we do not separate sharply between the physical world as it is and our theories about it, our discussion will be very confused. Perhaps we will never develop a completely accurate theory of the physical world. But if we do, if will be framed in a mathematical language, so we should think about what languages are available and create more if needed.

If we do not separate sharply between the physical world as it is and our theories about it, our discussion will be very confused. Perhaps we will never develop a completely accurate theory of the physical world. But if we do, if will be framed in a mathematical language, so we should think about what languages are available and create more if needed.

Thanks. I have a problem with this below you may want to give another shot at it although I understand that communicating through message boards is difficult especially in this area:

"If we do not separate sharply between the physical world as it is and our theories about it... "

Is it necessary that the physical world should be in a particular way or this is just an assumption to make our life easier?

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"If we do not separate sharply between the physical world as it is and our theories about it... "

Is it necessary that the physical world should be in a particular way or this is just an assumption to make our life easier?

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I do think we are having some communication problems....Even if the laws of the universe change through time, for example, there is still some way it is (i.e. changing, and changing in a particular way), and some description of it. So I can;t see any substantial assumption is saying that there is some way the universe is, not subject to our control (in the relevant sense) that we are trying to describe.

"Even if the laws of the universe change through time, for example, there is still some way it is (i.e. changing, and changing in a particular way), and some description of it."

The hidden assumption here is determinism. If this premise is true, then your statement is true otherwise it is false. I have trouble with the assumption of a "particular way" and of determinism. I think they both reflect some type of wish rather than a fact. Regardless of that, thank you.

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The hidden assumption here is determinism. If this premise is true, then your statement is true otherwise it is false. I have trouble with the assumption of a "particular way" and of determinism. I think they both reflect some type of wish rather than a fact. Regardless of that, thank you.

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Think of an indeterministic random walk. It is both the case that the particular details of any walk admit of a mathematical description (2 steps right, then one left, then three right, then four left...) and, in many cases, that some statistical characteristics of the walk can be predicted with high reliability. So indeterminism is not incompatible with mathematical description. Quantum theory is generally considered to be indeterministic, but still amenable to precise mathematical description.

Dear Tim,

I read your essay with great interest. I totally agree with you: "But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world." My high score.

I think only that we must first to consider the proto-structure of the Universum (matter) from the point of view of eternity ("sub specie aeternitatis"), that is, to carry out the ontological structure of matter in the proto-era, "time before times began". When we "grab" (understand) the primordial (ontological) structure of space, then we will understand the nature of time. Therefore, the fundamental physics we must move from the concept of "space-time", to the onto-topological concept**"space-matter-time"**. The primordial structure of matter determines the structure of the language in which Nature speaks to us, single language for mathematicians, physicists and poets , ie, language that contains all the meanings of the "LifeWorld"(E.Husserl). I invite you to read my essay .

Kind regards,

Vladimir

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I read your essay with great interest. I totally agree with you: "But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world." My high score.

I think only that we must first to consider the proto-structure of the Universum (matter) from the point of view of eternity ("sub specie aeternitatis"), that is, to carry out the ontological structure of matter in the proto-era, "time before times began". When we "grab" (understand) the primordial (ontological) structure of space, then we will understand the nature of time. Therefore, the fundamental physics we must move from the concept of "space-time", to the onto-topological concept

Kind regards,

Vladimir

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Dear Vladimir,

Certainly physics must deal with matter—a nice simple characterization of physics is the theory of matter in motion. What I have proposed here does not touch on that. Rather, it deals only with the "motion" part. Motion can be understood as the trajectory of an object through space-time, so then one question is how to characterize the structure (geometry) of space-time. That is what I have been working on. Putting the matter into the space-time arena is yet another problem (one would have to confront quantum theory). But I am just taking one step at a time—the step I have something new to offer.

Thanks for the comments.

TIm

Certainly physics must deal with matter—a nice simple characterization of physics is the theory of matter in motion. What I have proposed here does not touch on that. Rather, it deals only with the "motion" part. Motion can be understood as the trajectory of an object through space-time, so then one question is how to characterize the structure (geometry) of space-time. That is what I have been working on. Putting the matter into the space-time arena is yet another problem (one would have to confront quantum theory). But I am just taking one step at a time—the step I have something new to offer.

Thanks for the comments.

TIm

Dear Prof. Maudlin,

You make some interesting arguments about the nature of physics and mathematics, but it seems to me that the entire question of the “mysterious connection between physics and math” is misplaced. There is rather a simple explanation. Physics deals with how simple rules for relating real objects lead to more complex objects. Mathematics deals with how simple rules for relating abstract structures lead to more complex structures. So a common theme of underlying simplicity can guide them both. But there is no reason to assume that a given elegant mathematical model must ipso facto be represented in the real world.

My own essay addresses a somewhat different issue: ("Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory"

I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What’s more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

Alan Kadin

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You make some interesting arguments about the nature of physics and mathematics, but it seems to me that the entire question of the “mysterious connection between physics and math” is misplaced. There is rather a simple explanation. Physics deals with how simple rules for relating real objects lead to more complex objects. Mathematics deals with how simple rules for relating abstract structures lead to more complex structures. So a common theme of underlying simplicity can guide them both. But there is no reason to assume that a given elegant mathematical model must ipso facto be represented in the real world.

My own essay addresses a somewhat different issue: ("Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory"

I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What’s more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

Alan Kadin

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Dear Alan Kadin,

Thanks for your remarks. Of course, even if both physics and mathematics are concerned with situations where one wants to derive complex conclusions from relatively simple rules (and at one level of abstraction that is correct), it would not follow that the actual physical world behaves in a way well-desribed by a mathematical formalism. We would certainly like, for example, simple rules from which we could derive the weather a year from now, but it seems that the physics of weather just does not admit of such rules at all. So there is a question of which physical conditions must obtain for an effective mathematical description to be possible.

I am a bit puzzled by your description of your theory. If your theory is local, then, by Bell's theorem, it cannot predict violations of Bell inequality for experiments done at space-like separation. But these experiments have been done, and Bell's inequality is violated. So if the theory is as you have described it, we do not need to do further experiments with Stern-Gerlach magnets to check: relevant experiments already exist.

Regards,

Tim Maudlin

Thanks for your remarks. Of course, even if both physics and mathematics are concerned with situations where one wants to derive complex conclusions from relatively simple rules (and at one level of abstraction that is correct), it would not follow that the actual physical world behaves in a way well-desribed by a mathematical formalism. We would certainly like, for example, simple rules from which we could derive the weather a year from now, but it seems that the physics of weather just does not admit of such rules at all. So there is a question of which physical conditions must obtain for an effective mathematical description to be possible.

I am a bit puzzled by your description of your theory. If your theory is local, then, by Bell's theorem, it cannot predict violations of Bell inequality for experiments done at space-like separation. But these experiments have been done, and Bell's inequality is violated. So if the theory is as you have described it, we do not need to do further experiments with Stern-Gerlach magnets to check: relevant experiments already exist.

Regards,

Tim Maudlin

Dear Prof Maudlin,

From Wignar "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?"

His answer: It has to be admitted that we have no definite evidence that there is no such theory.

I am not sure of the detail regarding "Theory of Linear Structures", but I think there is room for multiple models, where details will either pass of fail the test of real physical experiments.

Regards and wondering if you have a internet link for more detail?

Ed

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From Wignar "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?"

His answer: It has to be admitted that we have no definite evidence that there is no such theory.

I am not sure of the detail regarding "Theory of Linear Structures", but I think there is room for multiple models, where details will either pass of fail the test of real physical experiments.

Regards and wondering if you have a internet link for more detail?

Ed

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Dear Ed,

The mathematical detail is spelled out in the book of mine I have cited, but is not online. Application of the mathematical to physical theories is the subject of a second volume that is being written now.

Regards,

Tim

The mathematical detail is spelled out in the book of mine I have cited, but is not online. Application of the mathematical to physical theories is the subject of a second volume that is being written now.

Regards,

Tim

Tim,

Thanks for your interesting and thought-provoking essay. I was wondering if you have applied your theory of linear structures to any of the discrete structures that have been proposed as candidate theories of quantum gravity, such as causal sets, spin foams, or causal dynamical triangulations?

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Thanks for your interesting and thought-provoking essay. I was wondering if you have applied your theory of linear structures to any of the discrete structures that have been proposed as candidate theories of quantum gravity, such as causal sets, spin foams, or causal dynamical triangulations?

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Hi Matt,

I have not tried to get the quantum-mechanical aspect of it, but I have done some work on using this to describe discrete Relativistic structures. So think of this as in the spirit of causal sets. I can get a simple discrete approximation to a 2-D Minkowski space-time and to a 3-D inflating space-time with horizons, and this is just from trying a few simple constructive rules for the Linear Structure and then analyzing the results. I have an idea for a general scheme for writing down constructive rules (both deterministic and stochastic) for generating Relativistic discrete Linear Structures, but there is a lot of work to do.

Just to give a taste of how this differs from causal sets, using the usual way that causal sets are generated no pair of events will be null related. But doing it my way, the entire space-time structure is built from null related events: it is all light-like in the foundations, as it were. I can also easily put in place constraints on the constructive rules that avoid some of the issues that come up for causal sets, which basically arise from the fact that the kind of graph they want to get is very much not a random graph.

The analytical advantage of a discrete space is that it comes already equipped with a natural measure—counting measure—but in the Relativistic case you have to be careful about what to count. I know that sounds cryptic, but it would take to long to explain properly...maybe we can talk about it sometime.

It may be that just being able to generate good discrete approximations to classical solutions in GR would yield clues about how to implement a fully quantum treatment, but that prospect is too far away now.

Cheers,

Tim

I have not tried to get the quantum-mechanical aspect of it, but I have done some work on using this to describe discrete Relativistic structures. So think of this as in the spirit of causal sets. I can get a simple discrete approximation to a 2-D Minkowski space-time and to a 3-D inflating space-time with horizons, and this is just from trying a few simple constructive rules for the Linear Structure and then analyzing the results. I have an idea for a general scheme for writing down constructive rules (both deterministic and stochastic) for generating Relativistic discrete Linear Structures, but there is a lot of work to do.

Just to give a taste of how this differs from causal sets, using the usual way that causal sets are generated no pair of events will be null related. But doing it my way, the entire space-time structure is built from null related events: it is all light-like in the foundations, as it were. I can also easily put in place constraints on the constructive rules that avoid some of the issues that come up for causal sets, which basically arise from the fact that the kind of graph they want to get is very much not a random graph.

The analytical advantage of a discrete space is that it comes already equipped with a natural measure—counting measure—but in the Relativistic case you have to be careful about what to count. I know that sounds cryptic, but it would take to long to explain properly...maybe we can talk about it sometime.

It may be that just being able to generate good discrete approximations to classical solutions in GR would yield clues about how to implement a fully quantum treatment, but that prospect is too far away now.

Cheers,

Tim

Sounds interesting. I will be at New Directions. Maybe we can discuss it there.

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This was a good essay with some interesting ideas such as the temporalization of sapce but ity is an idea that cannto be tested in a laboratory and abstract as it is it is pure speculation and increases the complexity of physical models and introduces more ambiguity. I think a theory like that presented in usenet years ago would be fircly attacked and the creator woudl be called names. What are any new predictions this temporalization ofefrs? Therefore, although I through the essay was good I think it makes undjustifuable claims that should not be made at the level of professional physics.

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"Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space"

Most theoreticians now believe that the spatialization of time (a consequence of Einstein's 1905 false constant-speed-of-light postulate) was wrong:

"And by making the clock's tick relative - what happens simultaneously for one observer might seem sequential to another - Einstein's theory of special relativity not only destroyed any notion of absolute time but made time equivalent to a dimension in space: the future is already out there waiting for us; we just can't see it until we get there. This view is a logical and metaphysical dead end, says Smolin."

"Was Einstein wrong? At least in his understanding of time, Smolin argues, the great theorist of relativity was dead wrong. What is worse, by firmly enshrining his error in scientific orthodoxy, Einstein trapped his successors in insoluble dilemmas..."

WHAT SCIENTIFIC IDEA IS READY FOR RETIREMENT? Steve Giddings: "Spacetime. Physics has always been regarded as playing out on an underlying stage of space and time. Special relativity joined these into spacetime... (...) The apparent need to retire classical spacetime as a fundamental concept is profound..."

Nima Arkani-Hamed 06:11 : "Almost all of us believe that space-time doesn't really exist, space-time is doomed and has to be replaced by some more primitive building blocks."

Pentcho Valev

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Most theoreticians now believe that the spatialization of time (a consequence of Einstein's 1905 false constant-speed-of-light postulate) was wrong:

"And by making the clock's tick relative - what happens simultaneously for one observer might seem sequential to another - Einstein's theory of special relativity not only destroyed any notion of absolute time but made time equivalent to a dimension in space: the future is already out there waiting for us; we just can't see it until we get there. This view is a logical and metaphysical dead end, says Smolin."

"Was Einstein wrong? At least in his understanding of time, Smolin argues, the great theorist of relativity was dead wrong. What is worse, by firmly enshrining his error in scientific orthodoxy, Einstein trapped his successors in insoluble dilemmas..."

WHAT SCIENTIFIC IDEA IS READY FOR RETIREMENT? Steve Giddings: "Spacetime. Physics has always been regarded as playing out on an underlying stage of space and time. Special relativity joined these into spacetime... (...) The apparent need to retire classical spacetime as a fundamental concept is profound..."

Nima Arkani-Hamed 06:11 : "Almost all of us believe that space-time doesn't really exist, space-time is doomed and has to be replaced by some more primitive building blocks."

Pentcho Valev

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Tim Maudlin: "The Relativistic account of space-time geometry makes the light-cone structure of space-time a fundamental part of its geometry. This, rather than the "constancy of the speed of light" lies at the heart of the theory."

This is wrong, Tim Maudlin - the constancy of the speed of light does lie at the heart of the theory, even if you want to hide it behind the light-cone structure of space-time. And since the speed of light is not constant (you know that don't you?), you will have to join Steve Giddings, Nima Arkani-Hamed and Lee Smolin in their rejection of Einstein's space-time.

Pentcho Valev

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This is wrong, Tim Maudlin - the constancy of the speed of light does lie at the heart of the theory, even if you want to hide it behind the light-cone structure of space-time. And since the speed of light is not constant (you know that don't you?), you will have to join Steve Giddings, Nima Arkani-Hamed and Lee Smolin in their rejection of Einstein's space-time.

Pentcho Valev

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Dear Prof Maudlin,

For your subquestion 1) „Which mathematical concepts seem naturally suited to describe features of the physical world, and what does their suitability Imply about the physical world?“

I suggest three main candidates for the mathematical concept:

bit (it was the subject of the competition FQXi 2013);

exp(x) (You know the unique features of this function);

Euler's identity. There are other useful functions, but less importance.

Suitable use of pervious can to describe features of the physical World.

What are your main candidates? If you agree with me, part of the solution can be found in my essay.

Best Regards,

Branko Zivlak

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For your subquestion 1) „Which mathematical concepts seem naturally suited to describe features of the physical world, and what does their suitability Imply about the physical world?“

I suggest three main candidates for the mathematical concept:

bit (it was the subject of the competition FQXi 2013);

exp(x) (You know the unique features of this function);

Euler's identity. There are other useful functions, but less importance.

Suitable use of pervious can to describe features of the physical World.

What are your main candidates? If you agree with me, part of the solution can be found in my essay.

Best Regards,

Branko Zivlak

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Tim,

I read your essay with a mixture of exhilaration and misgiving. Exhilaration for the sheer audacious brilliance of it, and misgiving that I have not yet introduced myself to your work. I plan to start correcting the latter in short order.

One comment: "This (metric - ed.) distance is just the minimal 1 length of a continuous path between the points. It can have an affine structure, which sorts continuous paths into straight and curved. It can have a differentiable structure, which distinguishes smooth curves from bent curves. But beneath all these, already presupposed by all of these, is the most basic geometrical structure: topological structure."

The straight line being a special case for the curve, an analytical "twoity" (LEJ Brouwer's word) guarantees curved structure of metric properties. In a 2006 conference paper I identified the complex plane structure that guarantees a counting function without appealing to the axiom of choice, with a physical definition of "time: n-dimension infinitely orientable metric on random, self-avoiding walk."

Looking forward to immersing myself in your research.

Tom

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I read your essay with a mixture of exhilaration and misgiving. Exhilaration for the sheer audacious brilliance of it, and misgiving that I have not yet introduced myself to your work. I plan to start correcting the latter in short order.

One comment: "This (metric - ed.) distance is just the minimal 1 length of a continuous path between the points. It can have an affine structure, which sorts continuous paths into straight and curved. It can have a differentiable structure, which distinguishes smooth curves from bent curves. But beneath all these, already presupposed by all of these, is the most basic geometrical structure: topological structure."

The straight line being a special case for the curve, an analytical "twoity" (LEJ Brouwer's word) guarantees curved structure of metric properties. In a 2006 conference paper I identified the complex plane structure that guarantees a counting function without appealing to the axiom of choice, with a physical definition of "time: n-dimension infinitely orientable metric on random, self-avoiding walk."

Looking forward to immersing myself in your research.

Tom

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Dear Tom,

Thanks for the comments. I was not aware of the Brouwer, and a quick look at some discussions shows that it will not be an easy thing to really understand. It is, of course, possible to describe the geometry of a space with enough structure to define the affine structure but not enough for a full metric. The so-called "Galilean" or "Neo-Newtonian" space-time is like this (if you try to use a standard full metric is it degenerate). This is particularly nice if one is trying to translate physical laws into a purely geometrical vocabulary. Newton's First Law, for example, becomes "The trajectory of a body is a straight line through space-time unless a force is put on it". The fundamental distinction between the affine and metrical structure also shows up when one demands, in General Relativity, that the metric be compatible with the connection on the tangent bundle.

Regards,

Tim

Thanks for the comments. I was not aware of the Brouwer, and a quick look at some discussions shows that it will not be an easy thing to really understand. It is, of course, possible to describe the geometry of a space with enough structure to define the affine structure but not enough for a full metric. The so-called "Galilean" or "Neo-Newtonian" space-time is like this (if you try to use a standard full metric is it degenerate). This is particularly nice if one is trying to translate physical laws into a purely geometrical vocabulary. Newton's First Law, for example, becomes "The trajectory of a body is a straight line through space-time unless a force is put on it". The fundamental distinction between the affine and metrical structure also shows up when one demands, in General Relativity, that the metric be compatible with the connection on the tangent bundle.

Regards,

Tim

Hi Tim,

I think a full metric description is, just as you imply -- native to point set topology, and not to affine space. There is an arithmetic theorem that any point maps simultaneously to any set of points provided it is far enough away. In reverse, this gives us the degenerate result of Galilean or Newtonian space. There is no time parameter.

Einstein, by introducing time by way of Minkowski space, may have hoped the point "far enough away" would avoid the singularity and instead found that expanding 4 dimension spacetime (by Hubble's result) places the singularity at every arbitrarily chosen point of 3-space. There will be singularities in general relativity. No point is far enough away to overcome the Poincare-Hopf theorem.

Best,

Tom

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I think a full metric description is, just as you imply -- native to point set topology, and not to affine space. There is an arithmetic theorem that any point maps simultaneously to any set of points provided it is far enough away. In reverse, this gives us the degenerate result of Galilean or Newtonian space. There is no time parameter.

Einstein, by introducing time by way of Minkowski space, may have hoped the point "far enough away" would avoid the singularity and instead found that expanding 4 dimension spacetime (by Hubble's result) places the singularity at every arbitrarily chosen point of 3-space. There will be singularities in general relativity. No point is far enough away to overcome the Poincare-Hopf theorem.

Best,

Tom

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Dear Tim,

I was very happy to find your essay here. I read it with pleasure and I like it so much. I am a mathematical physicist working in general relativity (singularities). Also, I started recently teaching a master class at the Faculty of Philosophy, together with the philosopher Iulian Toader, and we are using as main resource your book Philosophy of Physics. Space and Time, which we...

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I was very happy to find your essay here. I read it with pleasure and I like it so much. I am a mathematical physicist working in general relativity (singularities). Also, I started recently teaching a master class at the Faculty of Philosophy, together with the philosopher Iulian Toader, and we are using as main resource your book Philosophy of Physics. Space and Time, which we...

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Dear Crista,

Thanks so much for the comments. I am familiar with some work on pointless spaces (some of it older that the things coming from category theory) and I can only say that there seemed to be no obvious way to adapt this approach there. The basic problem is that the points in a 1-D open manifold are linearly ordered automatically, but while regions in a pointless space can include one another, it is hard to define the same sort of linear order if the basic structures are not 1-D (and hence constructed from things are are 0-D, i.e. points). This is not so say it is impossible, but that it is at least not obvious.

The language of Linear Structures is, as you appreciate, very flexible. That is good in one way (lots of possible geometries) and bad in another (you need strong constraints to narrow down to what you want). One interesting place to look is discrete spaces, when one can consider various constructive rules for generating a geometry and then analyze the character of the geometries that result. I have quickly found rules that give good discrete approximations to a 2-D flat space-time and the 3-D expanding space-time with a horizon structure. It will take more research to figure out exactly what features of the constructing rule control the outcome.

The idea is a bit like Causal sets, but the actual implementation is quite different. The most fundamental structure is light-like rather than time-like, and the Causal sets it is time-like.

Cheers,

Tim

Thanks so much for the comments. I am familiar with some work on pointless spaces (some of it older that the things coming from category theory) and I can only say that there seemed to be no obvious way to adapt this approach there. The basic problem is that the points in a 1-D open manifold are linearly ordered automatically, but while regions in a pointless space can include one another, it is hard to define the same sort of linear order if the basic structures are not 1-D (and hence constructed from things are are 0-D, i.e. points). This is not so say it is impossible, but that it is at least not obvious.

The language of Linear Structures is, as you appreciate, very flexible. That is good in one way (lots of possible geometries) and bad in another (you need strong constraints to narrow down to what you want). One interesting place to look is discrete spaces, when one can consider various constructive rules for generating a geometry and then analyze the character of the geometries that result. I have quickly found rules that give good discrete approximations to a 2-D flat space-time and the 3-D expanding space-time with a horizon structure. It will take more research to figure out exactly what features of the constructing rule control the outcome.

The idea is a bit like Causal sets, but the actual implementation is quite different. The most fundamental structure is light-like rather than time-like, and the Causal sets it is time-like.

Cheers,

Tim

Dear Tim,

I am happy to announce you that I found a natural way to supplement linear structures (directed null lines) with simple constraints to recover relativistic spacetime in n dimensions, without directly imposing to be locally IR^{n}. If you think it would be interesting, I can post it here, I don't think it will take me more than a couple of pages to explain it in detail.

Best wishes,

Cristi Stoica

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I am happy to announce you that I found a natural way to supplement linear structures (directed null lines) with simple constraints to recover relativistic spacetime in n dimensions, without directly imposing to be locally IR

Best wishes,

Cristi Stoica

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Dear Cristi,

Of course I would be very interested! Maybe you should send it directly in an e-mail so I can keep it more easily. My e-mail is twm3@nyu.edu.

Thanks!

Tim

Of course I would be very interested! Maybe you should send it directly in an e-mail so I can keep it more easily. My e-mail is twm3@nyu.edu.

Thanks!

Tim

Dear Tim Maudlin,

I really enjoyed the thread of your argument: simple arithmetic and geometry, based on our acquaintance with the physical world, leading to complex mathematical concepts, and these in-turn having some role in physics, and thus the Wigner's puzzle being solved. The whole thing would have been a smooth curve, but for your sudden jump: in explaining geometry, you jumped from...

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I really enjoyed the thread of your argument: simple arithmetic and geometry, based on our acquaintance with the physical world, leading to complex mathematical concepts, and these in-turn having some role in physics, and thus the Wigner's puzzle being solved. The whole thing would have been a smooth curve, but for your sudden jump: in explaining geometry, you jumped from...

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Dear Jose Koshy,

I agree that the motion of matter is described by some mathematical law, but the formulation of that law is beyond the scope of my present project. If we start with the general idea that physics is the theory of matter in motion, this presents two targets for an exact theory: matter and motion. The "motion" of a localized object seems to be best understood (and this is not how Newton would have conceptualized it) as a trajectory through space-time. So that leaves us with the problem of describing the geometry of the space-time, and the problem of understanding how the trajectories of objects are produced. You have to solve the first in order to even approach the second. That is, you need to understand the geometry of the space-time before you can begin to write down a dynamics. This project is then even more basic: not yet "what is the geometry of space-time?" but rather "what is the best mathematical language to use to describe the geometry of space-time?". If the fundamentals of Relativity are correct, this looks like an especially promising sort of language. The language is new, in the sense that (as far as I know) no one has written it down before. But at a basic level, I am a Platonist about mathematics, and in that sense nothing is really new.

Regards,

Tim

I agree that the motion of matter is described by some mathematical law, but the formulation of that law is beyond the scope of my present project. If we start with the general idea that physics is the theory of matter in motion, this presents two targets for an exact theory: matter and motion. The "motion" of a localized object seems to be best understood (and this is not how Newton would have conceptualized it) as a trajectory through space-time. So that leaves us with the problem of describing the geometry of the space-time, and the problem of understanding how the trajectories of objects are produced. You have to solve the first in order to even approach the second. That is, you need to understand the geometry of the space-time before you can begin to write down a dynamics. This project is then even more basic: not yet "what is the geometry of space-time?" but rather "what is the best mathematical language to use to describe the geometry of space-time?". If the fundamentals of Relativity are correct, this looks like an especially promising sort of language. The language is new, in the sense that (as far as I know) no one has written it down before. But at a basic level, I am a Platonist about mathematics, and in that sense nothing is really new.

Regards,

Tim

Dear Tim Maudlin,

Your stand is clear: if the fundamentals of (General) Relativity are correct, then there is four-dimensional space-time, and you are proposing a new language to describe the geometry of the space-time. I have just downloaded "New foundations for physical geometry" to know exactly what you referred as the project.

My argument is that physicists have not explored all the possibilities classical three-dimensional space offers. For example, the model proposed by me (refer finitenesstheory.com) views that motion at speed 'c' is the fundamental property of matter, and the reaction to this motion creates gravity. Consequently, the path of a body, in a classical 3-dimensional space, is bent by its own gravity. This can give similar results like that of GR.

From a cursory glance at the material downloaded, I get the impression that you consider going back to classical three-dimensional space is a retrograde step. However, I think that you will ultimately arrive at that conclusion.

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Your stand is clear: if the fundamentals of (General) Relativity are correct, then there is four-dimensional space-time, and you are proposing a new language to describe the geometry of the space-time. I have just downloaded "New foundations for physical geometry" to know exactly what you referred as the project.

My argument is that physicists have not explored all the possibilities classical three-dimensional space offers. For example, the model proposed by me (refer finitenesstheory.com) views that motion at speed 'c' is the fundamental property of matter, and the reaction to this motion creates gravity. Consequently, the path of a body, in a classical 3-dimensional space, is bent by its own gravity. This can give similar results like that of GR.

From a cursory glance at the material downloaded, I get the impression that you consider going back to classical three-dimensional space is a retrograde step. However, I think that you will ultimately arrive at that conclusion.

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Dear Jose Koshy,

I would not at all consider quite different space-tmes from GR as "retrograde": I am quite open to all sorts of possibilities. Here are two considerations. One is that if one rejects the idea that space-time is a continuum and uses a discrete space-time instead, the most natural way to implement that idea using this formalism results in the basic geometrical structure of space-time being all light like (null) paths. It would then follow automatically that any matter in continuous motion (i.e. following a continuous path) is always "traveling at the speed of light", and massive particles, such as electrons, must really engage in Zitterbewegung: rapid vibration. The other consequence of going to a discrete space-time is that the geometrical structure has a natural unique foliation, which can be considered a "backward" step to a more classical structure. (Not full Newtonian picture, with a single space persisting through time, but a structure with an intrinsic "simultaneity" built in, together with a light-cone structure.) Is that bad? It is actually the most straightforward way to be able to implement the non-locality implied by violations of Bell's inequality. So I am very open to these possibilities. As I said, in the first place this is just a new mathematical language. There are many things that can be described using it.

Reagrds,

Tim Maudlin

I would not at all consider quite different space-tmes from GR as "retrograde": I am quite open to all sorts of possibilities. Here are two considerations. One is that if one rejects the idea that space-time is a continuum and uses a discrete space-time instead, the most natural way to implement that idea using this formalism results in the basic geometrical structure of space-time being all light like (null) paths. It would then follow automatically that any matter in continuous motion (i.e. following a continuous path) is always "traveling at the speed of light", and massive particles, such as electrons, must really engage in Zitterbewegung: rapid vibration. The other consequence of going to a discrete space-time is that the geometrical structure has a natural unique foliation, which can be considered a "backward" step to a more classical structure. (Not full Newtonian picture, with a single space persisting through time, but a structure with an intrinsic "simultaneity" built in, together with a light-cone structure.) Is that bad? It is actually the most straightforward way to be able to implement the non-locality implied by violations of Bell's inequality. So I am very open to these possibilities. As I said, in the first place this is just a new mathematical language. There are many things that can be described using it.

Reagrds,

Tim Maudlin

Dear Tim Mauldin

An enjoyable essay to read. I agree that “one could easily write a companion paper to Wigner’s called “The Unreasonable Relevance of Some Branches of Mathematics to Other Branches””. For me there are 2 places where this particular unreasonableness of maths transfers over to physics in a sort of pincer movement that constrains physics and potentially your proposed...

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An enjoyable essay to read. I agree that “one could easily write a companion paper to Wigner’s called “The Unreasonable Relevance of Some Branches of Mathematics to Other Branches””. For me there are 2 places where this particular unreasonableness of maths transfers over to physics in a sort of pincer movement that constrains physics and potentially your proposed...

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Dear Michael,

I do not feel confident that I am in command of the technical details of your analysis, but let me say at least this. The sense in which any new theory must "reproduce the descriptions of existing theory" is obviously a matter of approximation, not exact derivation. That is, all experimental data come with error bars, and so recovering the predictive success of present theories (such as quantum theory or the General Theory of Relativity) requires matching their predictions where they have actually been tested to within the tolerance of the errors. If one were to demand a higher degree of match than this, then the new description would of course be "lost". But the expectation is that the new description will deviate in its predictions from the present theory, but only to a small degree and only in certain circumstances. What that degree is, and which circumstances are relevant depends on the theory itself.

Regards,

Tim Maudlin

I do not feel confident that I am in command of the technical details of your analysis, but let me say at least this. The sense in which any new theory must "reproduce the descriptions of existing theory" is obviously a matter of approximation, not exact derivation. That is, all experimental data come with error bars, and so recovering the predictive success of present theories (such as quantum theory or the General Theory of Relativity) requires matching their predictions where they have actually been tested to within the tolerance of the errors. If one were to demand a higher degree of match than this, then the new description would of course be "lost". But the expectation is that the new description will deviate in its predictions from the present theory, but only to a small degree and only in certain circumstances. What that degree is, and which circumstances are relevant depends on the theory itself.

Regards,

Tim Maudlin

Dear Professor Maudlin,

this is a very interesting essay and a very interesting program of research. It reminds me very much of the causal set research program of Rafael Sorkin. In particular, like causal sets it is a beautiful and austere way of looking at spacetime structure, but perhaps a bit too austere for my taste. Below I append the sort of questions that I usually ask about the causal set program which I think also apply to your program. If you have time to reply to any of these questions, that would be very much appreciated.

--David Garfinkle

(1) Your method gives a conformal structure, which in the case of a spacetime is equivalent to the conformal class of the metric. There is then a standard result that also giving the volume element will determine the metric. Is there a simple and natural way of specifying a volume element within your formalism?

(2) Presumably a model for a spacetime within your formalism would be a set with your structure which also (exactly or approximately) admits a differential structure (thus making it a manifold) and a metric. But most of the sets with your structure will not be manifolds, even in an approximate sense. Is their a way within your formalism of figuring out which sets are manifolds? (or approximately manifolds).

(3) I admire the austere beauty of your approach. But I'm pretty happy with the standard notion of spacetime as a manifold with a metric and with the usual definition of a manifold as a topological space with an atlas. What do I lose by not adopting your approach?

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this is a very interesting essay and a very interesting program of research. It reminds me very much of the causal set research program of Rafael Sorkin. In particular, like causal sets it is a beautiful and austere way of looking at spacetime structure, but perhaps a bit too austere for my taste. Below I append the sort of questions that I usually ask about the causal set program which I think also apply to your program. If you have time to reply to any of these questions, that would be very much appreciated.

--David Garfinkle

(1) Your method gives a conformal structure, which in the case of a spacetime is equivalent to the conformal class of the metric. There is then a standard result that also giving the volume element will determine the metric. Is there a simple and natural way of specifying a volume element within your formalism?

(2) Presumably a model for a spacetime within your formalism would be a set with your structure which also (exactly or approximately) admits a differential structure (thus making it a manifold) and a metric. But most of the sets with your structure will not be manifolds, even in an approximate sense. Is their a way within your formalism of figuring out which sets are manifolds? (or approximately manifolds).

(3) I admire the austere beauty of your approach. But I'm pretty happy with the standard notion of spacetime as a manifold with a metric and with the usual definition of a manifold as a topological space with an atlas. What do I lose by not adopting your approach?

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Dear Professor Garfinkle,

I completely understand seeing a similarity between this approach and causal sets, but at the level of implementation and exact detail they are quite different. So, for example, in the causal set approach there is essentially zero chance for any pair of events to be null related, and in this approach to a discrete structure all the fundamental space-time geometry...

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I completely understand seeing a similarity between this approach and causal sets, but at the level of implementation and exact detail they are quite different. So, for example, in the causal set approach there is essentially zero chance for any pair of events to be null related, and in this approach to a discrete structure all the fundamental space-time geometry...

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the central point here is :(Q) what properties mus physical reality have for math to be applicable? As the author knows the brings in some traditional problems of scientific realism. In the simple cases treated first we know reality and then apply integers or geometric forms. In the difficult cases we know physical reality through the way successful physical theories represent reality. Here Q i still applicable but more difficult to answer. You eventually sharpen Q to: What physical features space or space-time must have to be represented by the topology of an open set? This leads into your development of a theory of linear structures. I don't feel qualified to comment on that, though it looks good. I tend t think of open sets as a mathematical trick with no direct physical significance. Maybe the development of your theory will chang that.

Ed MacKinnon

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Ed MacKinnon

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Dear Tim Maudlin,

You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.

Yes I agree with you and thats why I have propounded...

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You have mentioned that for mathematics to be used as the language of physics, physical world has that sort of structure to be represented mathematically? That depends on the mathematical language being used Physical characteristics are required for mathematical structures to describe a physical situation.

Yes I agree with you and thats why I have propounded...

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Dear Pankaj Mani,

Thank you for the comments. The use of complex numbers in quantum theory is a very interesting case, which needs a lot of discussion. My own work here is just on space-time structure, so does not touch on quantum theory directly. But I think it may help to recall that time-revesal is implemented in quantum formalism by taking the complex conjugate of the wave function. This immediately suggests a connection between the use of complex numbers and the temporal structure, indeed a connection with the direction of time.

It is harder to deal with quantum theory because there is no agreement at all about just what physical entities the theory is committed to, particularly what Bell called the "local beables" of the theory. The observable behavior of laboratory apparatus should be determined by the behavior of these local beables at microscopic scale. If you don't even know what these are, then interpreting the significance of the mathematical apparatus becomes essentially impossible. Pure space-time theory is a bit more straightforward.

Regards,

Tim Maudlin

Thank you for the comments. The use of complex numbers in quantum theory is a very interesting case, which needs a lot of discussion. My own work here is just on space-time structure, so does not touch on quantum theory directly. But I think it may help to recall that time-revesal is implemented in quantum formalism by taking the complex conjugate of the wave function. This immediately suggests a connection between the use of complex numbers and the temporal structure, indeed a connection with the direction of time.

It is harder to deal with quantum theory because there is no agreement at all about just what physical entities the theory is committed to, particularly what Bell called the "local beables" of the theory. The observable behavior of laboratory apparatus should be determined by the behavior of these local beables at microscopic scale. If you don't even know what these are, then interpreting the significance of the mathematical apparatus becomes essentially impossible. Pure space-time theory is a bit more straightforward.

Regards,

Tim Maudlin

Dear Tim,

Much fascinated by your work, ‘The Theory of Linear Structures’. I think you may find some of the interesting applications of your work at a paradigm used for the comparative analysis in my essay, ‘ Before the Primordial Geometric origin: The Mysterious connection between Physics and Mathematics’. Hope you will enjoy in reading.

With best wishes,

Jayakar

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Much fascinated by your work, ‘The Theory of Linear Structures’. I think you may find some of the interesting applications of your work at a paradigm used for the comparative analysis in my essay, ‘ Before the Primordial Geometric origin: The Mysterious connection between Physics and Mathematics’. Hope you will enjoy in reading.

With best wishes,

Jayakar

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Dear Tim,

Concerning you criticisms in my Essay page, some clarifications could be needed. For rotating frame in my Essay I mean the frame in which the observer sees the detector at rest (the absorber orbits around the source). Clearly, in that frame photons propagate in the radial direction. You are of course correct in highlighting that Equivalence Principle has local behavior. On the other hand, rotating frames generate the centrifuge acceleration in the radial direction cited above, which, in turn, defines a locally accelerated frame. Thus, it seems to me that the application of Equivalence Principle is completely legitimate. I also stress that the use of the Equivalence Principle in rotating frames in general and in the Mössbauer rotor experiment in particular has a long, more than fifty-year-old, history. In the paper of Kündig, i.e. ref. [3] in my Essay, which is dated 1963, one reads verbatim: "when the experiment is analyzed in a reference frame K attached to the accelerate observer, the problem could be treated [7] by the principle of equivalence of the general theory of relativity". Reference [7] in the paper of Kündig is the historical book of Pauli on the theory of relativity dated 1958. Thus, it seems that you were wrong in those criticisms. Here the key point is not the viability of the Equivalence Principle in treating this problem, but the issue that previous literature did not take into due account clock synchronization.

I will read, comment and score your Essay soon. I wish you best luck in the Contest.

Cheers, Ch.

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Concerning you criticisms in my Essay page, some clarifications could be needed. For rotating frame in my Essay I mean the frame in which the observer sees the detector at rest (the absorber orbits around the source). Clearly, in that frame photons propagate in the radial direction. You are of course correct in highlighting that Equivalence Principle has local behavior. On the other hand, rotating frames generate the centrifuge acceleration in the radial direction cited above, which, in turn, defines a locally accelerated frame. Thus, it seems to me that the application of Equivalence Principle is completely legitimate. I also stress that the use of the Equivalence Principle in rotating frames in general and in the Mössbauer rotor experiment in particular has a long, more than fifty-year-old, history. In the paper of Kündig, i.e. ref. [3] in my Essay, which is dated 1963, one reads verbatim: "when the experiment is analyzed in a reference frame K attached to the accelerate observer, the problem could be treated [7] by the principle of equivalence of the general theory of relativity". Reference [7] in the paper of Kündig is the historical book of Pauli on the theory of relativity dated 1958. Thus, it seems that you were wrong in those criticisms. Here the key point is not the viability of the Equivalence Principle in treating this problem, but the issue that previous literature did not take into due account clock synchronization.

I will read, comment and score your Essay soon. I wish you best luck in the Contest.

Cheers, Ch.

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Dear Tim,

I was reading your excellent essay with growing interest as you were getting closer and closer to the geometry of spacetime. You argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to spacetime and you propose an alternative geometrical language. That is really promising as I am just looking for such languages. You finally claim that...

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I was reading your excellent essay with growing interest as you were getting closer and closer to the geometry of spacetime. You argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to spacetime and you propose an alternative geometrical language. That is really promising as I am just looking for such languages. You finally claim that...

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Dear Jacek,

As I mentioned in the appendix, there are topologies that cannot be recovered from any underlying Directed Linear Structure, but I am morally certain that these have no bearing at all on Thurston's conjecture, which deals with manifolds. I see no reason at all to doubt that every one of the manifolds that the conjecture deals with is generated by at least one (and in fact infinitely many) distinct Directed Linear Structures. (Indeed, by infinitely many Linear Structures, with no direction). And the way to implement the analog of surgery theory in the Theory of Linear Structures is completely straightforward: one specifies how two Linear Structures are to be combined by specifying which lines in one are continuations of lines in the other.

As for the profit: the Theory of Linear Structures describes geometrical structure in finer detail than standard topology: that is why many distinct Linear Structures typically can generate the same standard topology.

I'm not sure what you mean when you say that one cannot always assign a geometry to the whole space. I think you must be using that term in a non-obvious way. As you say, the theorem is about decomposing closed 3-manifolds into a set of pieces, each with a specified geometry. But there is obviously a geometry (in the sense of a topological and differentiable structure) assigned to all of these objects: that's why they count as 3-manifolds.

Regards,

Tim

As I mentioned in the appendix, there are topologies that cannot be recovered from any underlying Directed Linear Structure, but I am morally certain that these have no bearing at all on Thurston's conjecture, which deals with manifolds. I see no reason at all to doubt that every one of the manifolds that the conjecture deals with is generated by at least one (and in fact infinitely many) distinct Directed Linear Structures. (Indeed, by infinitely many Linear Structures, with no direction). And the way to implement the analog of surgery theory in the Theory of Linear Structures is completely straightforward: one specifies how two Linear Structures are to be combined by specifying which lines in one are continuations of lines in the other.

As for the profit: the Theory of Linear Structures describes geometrical structure in finer detail than standard topology: that is why many distinct Linear Structures typically can generate the same standard topology.

I'm not sure what you mean when you say that one cannot always assign a geometry to the whole space. I think you must be using that term in a non-obvious way. As you say, the theorem is about decomposing closed 3-manifolds into a set of pieces, each with a specified geometry. But there is obviously a geometry (in the sense of a topological and differentiable structure) assigned to all of these objects: that's why they count as 3-manifolds.

Regards,

Tim

Thank you Tim for inspiration and clarifying as the TLS is new for me. I do not feel it yet and I have to catch up.

When I say that one cannot always assign a (single) geometry to the whole space I mean this is not possible in three dimensions. This is specific feature of three dimensions pointed out in the geometrization conjecture. Differently, for two-dimensional surfaces you can freely assign a single geometry to a whole space. It was really not clearly stated.

Best regards,

Jacek

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When I say that one cannot always assign a (single) geometry to the whole space I mean this is not possible in three dimensions. This is specific feature of three dimensions pointed out in the geometrization conjecture. Differently, for two-dimensional surfaces you can freely assign a single geometry to a whole space. It was really not clearly stated.

Best regards,

Jacek

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Nicely argued essay, Tim!

I believe your idea that time (motion) imposes linear ordering on space is fundamental. I suppose you know that the idea was fundamental to Newton’s fluxions. And you have convinced me to look at your book on “The Theory of Linear Structures.”

In regard to geometry, I submit that your arguments may benefit from the Geometric Algebra mentioned in my essay. It may help you with the notion of areas defined by linear ordering of line segments, and volumes defined by linear ordering of areas.

Respectfully….David Hestenes

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I believe your idea that time (motion) imposes linear ordering on space is fundamental. I suppose you know that the idea was fundamental to Newton’s fluxions. And you have convinced me to look at your book on “The Theory of Linear Structures.”

In regard to geometry, I submit that your arguments may benefit from the Geometric Algebra mentioned in my essay. It may help you with the notion of areas defined by linear ordering of line segments, and volumes defined by linear ordering of areas.

Respectfully….David Hestenes

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Thanks David! I have always been an admirer of Geometric Algebra...one of my students (Doug Kutach) became much more adept at it than I did. I hope to be able to understand it more deeply. And on a different note, I have recently been looking into Zitter theory. It looks like if one uses Linear Structures to model discrete Relativistic space-times something like Zitter theory must be the right picture of matter. (The fundamental structure of the discrete models is all light-like, so all particles have to follow light-like paths at micro scale.)

Cheers,

Tim

Cheers,

Tim

Dear Tim,

just a quick first reaction to your enjoyable text. We are all very pleased to live in a physical world not completely described by fluid mechanics, but in one where chairs, tables, even living bodies are possible. The step from the physics of this type of world to the mathematics of natural numbers is short and reasonable. But in this reasoning (which I am not questioning) one is actually going form physics to math, from object to description, from territory to map. The reversed scenario - from math to physics - is also interesting, and more challenging. For example, the Mathematical Universe Hypothesis (MUH) does that: it puts math at the roots of the physical world, which would neatly explain why math is*also* good at *describing* the physics: math—(is)-->physics--(described-by)—>math. However, MUH does not specifically address the key question of why there are objects (thus natural numbers) rather than just fluids, a circumstance that scientists should also put in the box of ‘unreasonable’ facts about the universe (nice topic for next year Contest...).

In my opinion, the most convincing explanation for the emergence of distinguishable ‘objects’ in our universe, thus of natural numbers for their mathematical description, is, currently, the one that attributes a fundamentally*algorithmic* nature to the dynamics of spacetime, or of whichever discrete structure one figures sitting at the bottom of the universal architecture. We have today plenty of experimental evidence for the ‘miraculous’ emergence of distinguishable-denumerable ‘objects’ - object/background patterns - from the computations of even the simplest programs.

Best regards

Tommaso

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just a quick first reaction to your enjoyable text. We are all very pleased to live in a physical world not completely described by fluid mechanics, but in one where chairs, tables, even living bodies are possible. The step from the physics of this type of world to the mathematics of natural numbers is short and reasonable. But in this reasoning (which I am not questioning) one is actually going form physics to math, from object to description, from territory to map. The reversed scenario - from math to physics - is also interesting, and more challenging. For example, the Mathematical Universe Hypothesis (MUH) does that: it puts math at the roots of the physical world, which would neatly explain why math is

In my opinion, the most convincing explanation for the emergence of distinguishable ‘objects’ in our universe, thus of natural numbers for their mathematical description, is, currently, the one that attributes a fundamentally

Best regards

Tommaso

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Hi Tim,

After reading Tommaso's comment (but before I've read your essay), I wanted to chime in to say that the piece he describes as missing is precisely what I found almost 30 years ago, and touch on in my essay. The attached JPEG shows explicitly the quantum hydrodynamic analog within the Mandelbrot Set, showing where objectified forms appear just past the primary Misiurewicz points. This image is my Mandelbrot Butterfly figure, everted about (-1, 0i) such that concentric circles are laid flat.

When I sent this image to John Bush, he replied that he found it 'quite interesting,' which I suspect is because the analogy with quantum hydrodynamics is pretty obvious. This research is still a work in progress. But if it turns out there is a robust connection, where M reveals the process of pinch-off and nucleation by which fluids form droplets - this raises the issue that Tommaso raises above. How could this be, unless the physical reality flows directly from the Math - rather than the other way around?

All the Best,

Jonathan

attachments: Plateau.jpg

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After reading Tommaso's comment (but before I've read your essay), I wanted to chime in to say that the piece he describes as missing is precisely what I found almost 30 years ago, and touch on in my essay. The attached JPEG shows explicitly the quantum hydrodynamic analog within the Mandelbrot Set, showing where objectified forms appear just past the primary Misiurewicz points. This image is my Mandelbrot Butterfly figure, everted about (-1, 0i) such that concentric circles are laid flat.

When I sent this image to John Bush, he replied that he found it 'quite interesting,' which I suspect is because the analogy with quantum hydrodynamics is pretty obvious. This research is still a work in progress. But if it turns out there is a robust connection, where M reveals the process of pinch-off and nucleation by which fluids form droplets - this raises the issue that Tommaso raises above. How could this be, unless the physical reality flows directly from the Math - rather than the other way around?

All the Best,

Jonathan

attachments: Plateau.jpg

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... and another point.

You are against attributing an abstract mathematical status to democritean atoms: for being mathematically tractable (e.g. counted), they need some physical properties, that european mountains to not possess. You conclude that attributing a mathematical structure to physical items is not the same as postulating that they*are* mathematical entities.

I do not know which precise physical properties can be attributed to democritean atoms (probably not color, volume, spin…), but let us consider an atomic event, or atom of spacetime, as conceived in a causal set, a model mentioned higher up in this blog. These are points, with no other attributes than those you assign to the ideal mathematical/geometrical point. They are countable - crucially, for recovering volume information - and yet totally featureless. I would say that their nature (their ontological status?) is mathematical, not physical. And yet, when collected in a causal set, or a superposition of these, they are conjectured to originate, by emergence, just about Everything - I mean the physical Everything.

I find irresistible the argument that the deeper you go in magnifying the 'fabric of the cosmos’ the more the familiar physical properties we are used to recognize tend to vanish. Brought to the limit, this means that physical reality pulverises into mathematics, or ‘baggage-free’, purely abstract objects.

I guess you disagree with this, If I understand your points correctly…

Sorry for the length

Tommaso

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You are against attributing an abstract mathematical status to democritean atoms: for being mathematically tractable (e.g. counted), they need some physical properties, that european mountains to not possess. You conclude that attributing a mathematical structure to physical items is not the same as postulating that they

I do not know which precise physical properties can be attributed to democritean atoms (probably not color, volume, spin…), but let us consider an atomic event, or atom of spacetime, as conceived in a causal set, a model mentioned higher up in this blog. These are points, with no other attributes than those you assign to the ideal mathematical/geometrical point. They are countable - crucially, for recovering volume information - and yet totally featureless. I would say that their nature (their ontological status?) is mathematical, not physical. And yet, when collected in a causal set, or a superposition of these, they are conjectured to originate, by emergence, just about Everything - I mean the physical Everything.

I find irresistible the argument that the deeper you go in magnifying the 'fabric of the cosmos’ the more the familiar physical properties we are used to recognize tend to vanish. Brought to the limit, this means that physical reality pulverises into mathematics, or ‘baggage-free’, purely abstract objects.

I guess you disagree with this, If I understand your points correctly…

Sorry for the length

Tommaso

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Dear Tommaso,

I am a Platonist about mathematics, in the sense that I think there are objective facts about abstract mathematical objects, including facts that lie beyond our abilities to prove. (For example, Goldbach's conjecture is certainly either true or false, and if it is true we may never be able to prove it.) So there are all of these non-material mathematical structures, whose existence is independent of the physical world. (The physical world could not come out one way that makes Goldbach's conjecture true and another way that makes it false.) As I understand Tegmark's hypothesis, every single one of these abstract mathematical structures is a concrete physical world. I think that there are very, very severe problems of different sorts with that hypothesis. One is that the vast majority of possible mathematical structures are not regular enough to be described simply (think of all possible sequences of integers: for most there is no compact way to specify it). So if all mathematical structures are physical, most physical worlds are not compactly describable. And it would be almost a miracle that ours is.

That is a completely different matter than the one about algorithmic dynamics. Here I think we agree: indeed, relatively few structures can be generated by a compact algorithm, just a relatively few sequences can be generated by simple rules. The search for such an algorithm is a form of the search for simple laws. And the simplicity of the laws should explain the comprehensibility and predictability of the physical world.

Regards,

Tim

I am a Platonist about mathematics, in the sense that I think there are objective facts about abstract mathematical objects, including facts that lie beyond our abilities to prove. (For example, Goldbach's conjecture is certainly either true or false, and if it is true we may never be able to prove it.) So there are all of these non-material mathematical structures, whose existence is independent of the physical world. (The physical world could not come out one way that makes Goldbach's conjecture true and another way that makes it false.) As I understand Tegmark's hypothesis, every single one of these abstract mathematical structures is a concrete physical world. I think that there are very, very severe problems of different sorts with that hypothesis. One is that the vast majority of possible mathematical structures are not regular enough to be described simply (think of all possible sequences of integers: for most there is no compact way to specify it). So if all mathematical structures are physical, most physical worlds are not compactly describable. And it would be almost a miracle that ours is.

That is a completely different matter than the one about algorithmic dynamics. Here I think we agree: indeed, relatively few structures can be generated by a compact algorithm, just a relatively few sequences can be generated by simple rules. The search for such an algorithm is a form of the search for simple laws. And the simplicity of the laws should explain the comprehensibility and predictability of the physical world.

Regards,

Tim

Dear Dr. Maudlin:

I have downloaded a fair number of FQXi-2015 Contest Essays, and tried to read through as many as I can manage. Needless to say that my understanding of the essays is based on the framework I used to view them, and that framework is described in my essay http://fqxi.org/community/forum/topic/2456 .

Simply put, I view the world “analogically,” as contextually sensitive set of duals: i.e. I frame Wigner’s Refrain of mathematics and physics as freedom and determinism (among others I can choose) and then try to understand Dr. Maudlin’s:

(1) “Wigner’s question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh.”

(2) “Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation. The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world.”

Given my formulation of Wigner’s Thesis, there is nothing that I can disagree with Dr. Maudlin’s views as expressed in the two paragraphs above, but I like to know how the “meshing” might be accomplished in the project.

Regards,

Than Tin

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I have downloaded a fair number of FQXi-2015 Contest Essays, and tried to read through as many as I can manage. Needless to say that my understanding of the essays is based on the framework I used to view them, and that framework is described in my essay http://fqxi.org/community/forum/topic/2456 .

Simply put, I view the world “analogically,” as contextually sensitive set of duals: i.e. I frame Wigner’s Refrain of mathematics and physics as freedom and determinism (among others I can choose) and then try to understand Dr. Maudlin’s:

(1) “Wigner’s question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh.”

(2) “Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation. The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world.”

Given my formulation of Wigner’s Thesis, there is nothing that I can disagree with Dr. Maudlin’s views as expressed in the two paragraphs above, but I like to know how the “meshing” might be accomplished in the project.

Regards,

Than Tin

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Hi Tim,

Nice essay. I will have to read more about your Theory of Linear Structures at some point. It seems to have some similarity to what Kevin Knuth has done with posets, and with some of the things I'm doing right now with information orders on domains (building on the domain work of Keye Martin -- not sure if you are familiar with any of it).

I think you also hit on something interesting in regard to this idea of counting and how it relates to a physical ontology. One could argue that, even if the universe is entirely continuous, our ability to measure it to arbitrary accuracy is necessarily discrete and thus the integers match up well with that discreteness (which interestingly links back to a previous FQXi essay contest). Just a thought.

Anyway, I nevertheless must admit that I didn't find your argument convincing in general. It seems to miss some subtleties. Perhaps these subtleties are addressed in your larger work on the topic, however. For instance, I disagree with you on a key point: I do think that how different branches of mathematics relate to one another, has a direct bearing on how mathematics relates to the physical universe. How could it not? If you are familiar with category theory or topos theory, think about how such theories describe both mathematics and physics and their inter-relationships.

I had a few minor quibbles as well. In the example you gave of a universe describable entirely via fluid mechanics and dynamics, you would still be faced with the distinction of "something" versus "nothing" which maps quite naturally to 1 and 0 respectively. Integers are an elementary extrapolation from there.

I also am not particularly awed by the fact that results in semi-stable elliptical curves were used to prove Fermat's Last Theorem. While I am not deeply familiar with the details of Wiles' proof, in some sense both elliptic curves and Fermat's Last Theorem deal, on some level, with polynomials. Certainly the connection is not obvious, but neither is it all that shocking, at least to me.

Cheers,

Ian

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Nice essay. I will have to read more about your Theory of Linear Structures at some point. It seems to have some similarity to what Kevin Knuth has done with posets, and with some of the things I'm doing right now with information orders on domains (building on the domain work of Keye Martin -- not sure if you are familiar with any of it).

I think you also hit on something interesting in regard to this idea of counting and how it relates to a physical ontology. One could argue that, even if the universe is entirely continuous, our ability to measure it to arbitrary accuracy is necessarily discrete and thus the integers match up well with that discreteness (which interestingly links back to a previous FQXi essay contest). Just a thought.

Anyway, I nevertheless must admit that I didn't find your argument convincing in general. It seems to miss some subtleties. Perhaps these subtleties are addressed in your larger work on the topic, however. For instance, I disagree with you on a key point: I do think that how different branches of mathematics relate to one another, has a direct bearing on how mathematics relates to the physical universe. How could it not? If you are familiar with category theory or topos theory, think about how such theories describe both mathematics and physics and their inter-relationships.

I had a few minor quibbles as well. In the example you gave of a universe describable entirely via fluid mechanics and dynamics, you would still be faced with the distinction of "something" versus "nothing" which maps quite naturally to 1 and 0 respectively. Integers are an elementary extrapolation from there.

I also am not particularly awed by the fact that results in semi-stable elliptical curves were used to prove Fermat's Last Theorem. While I am not deeply familiar with the details of Wiles' proof, in some sense both elliptic curves and Fermat's Last Theorem deal, on some level, with polynomials. Certainly the connection is not obvious, but neither is it all that shocking, at least to me.

Cheers,

Ian

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Dear Ian,

Thanks for the comments. Let me try to address some of them.

There are surface similarities to how one treats discrete spaces or space-time using this formalism and what is done in Causal Set theory (which also uses posets), but the actual details turn out to be quite different. Of course, I did not have the space to go into that here, and it is not even in the book that is out, which deals solely with the math. The second volume will apply the math to physics, and it will be done there.

I friend of mine pointed out that for another reason there are countable things even in a fluid mechanics continuum: there can be discrete vortices. (There will be problems counting when they merge, but still they can be stable and discrete over long periods.) So the claims about fluid mechanics is too strong.

I think you misinterpreted the claim about the bearing of different branches of math on one another. Of course that has implications for the connection between mathematics and the world! My point was that if one branch has unproblematic relation to the physics, then any other mathematical structure which connects to the unproblematic one will inherit a comprehensible bearing on physics. In this case, I said that Wigner's problem is solved without remainder. I just wanted to separate puzzlement about why one branch of pure math bears on another from the question of why any math bears on physics.

The example of Fermat was just illustrative: maybe the connection is not so obscure. Like you, I do not know the details. Take the Moonshine conjectures then: certainly mathematicians were surprised about the connections between group theory and the Fourier expansion there. But if the physics were using the group theory in some obvious way, the purely mathematical connection would make the Fourier expansion relevant to study.

Cheers,

Tim

Thanks for the comments. Let me try to address some of them.

There are surface similarities to how one treats discrete spaces or space-time using this formalism and what is done in Causal Set theory (which also uses posets), but the actual details turn out to be quite different. Of course, I did not have the space to go into that here, and it is not even in the book that is out, which deals solely with the math. The second volume will apply the math to physics, and it will be done there.

I friend of mine pointed out that for another reason there are countable things even in a fluid mechanics continuum: there can be discrete vortices. (There will be problems counting when they merge, but still they can be stable and discrete over long periods.) So the claims about fluid mechanics is too strong.

I think you misinterpreted the claim about the bearing of different branches of math on one another. Of course that has implications for the connection between mathematics and the world! My point was that if one branch has unproblematic relation to the physics, then any other mathematical structure which connects to the unproblematic one will inherit a comprehensible bearing on physics. In this case, I said that Wigner's problem is solved without remainder. I just wanted to separate puzzlement about why one branch of pure math bears on another from the question of why any math bears on physics.

The example of Fermat was just illustrative: maybe the connection is not so obscure. Like you, I do not know the details. Take the Moonshine conjectures then: certainly mathematicians were surprised about the connections between group theory and the Fourier expansion there. But if the physics were using the group theory in some obvious way, the purely mathematical connection would make the Fourier expansion relevant to study.

Cheers,

Tim

Dear Prof. Maudlin.

Despite some interesting ideas, you paper pressuposes what it was supposed to explain. As I see it, there is nothing, absolutely nothing intrinsically mathematical in brute nature. Take for instance number. Given any, any!, amount of objects, no matter how sharply distinct one from another, from a certain perspective, there's no number naturally attached to it independently of a unit determination, or, which is the same, a concept which tells us what is it that we are numbering. So, numbering is a conceptual operation and concepts are creatures of ours. In my paper ("Mathematics, the Oracle of Physics") I approached the question of the applicability of mathematics in science from a transcendental perspective. Since nature "out there" has nothing intrinsically mathematical about it, how come that mathematics has anything to do with our theory of nature? From my point of view, the answer to this question requires showing how by a series of constituting acts a suitable mathematical representation of nature is constituted from brute sensorial data. Once this is done the applicability of mathematics in physics is, as I've argued, just an instance of the applicability of mathematics in mathematics itself (in your paper you explicitly reject this identification). In short, I don't think your perspective is radical enough from a truly philosophical perspective. You take too much for granted and embrace too many idees recues. Thank you! Best! Jairo Jose da Silva

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Despite some interesting ideas, you paper pressuposes what it was supposed to explain. As I see it, there is nothing, absolutely nothing intrinsically mathematical in brute nature. Take for instance number. Given any, any!, amount of objects, no matter how sharply distinct one from another, from a certain perspective, there's no number naturally attached to it independently of a unit determination, or, which is the same, a concept which tells us what is it that we are numbering. So, numbering is a conceptual operation and concepts are creatures of ours. In my paper ("Mathematics, the Oracle of Physics") I approached the question of the applicability of mathematics in science from a transcendental perspective. Since nature "out there" has nothing intrinsically mathematical about it, how come that mathematics has anything to do with our theory of nature? From my point of view, the answer to this question requires showing how by a series of constituting acts a suitable mathematical representation of nature is constituted from brute sensorial data. Once this is done the applicability of mathematics in physics is, as I've argued, just an instance of the applicability of mathematics in mathematics itself (in your paper you explicitly reject this identification). In short, I don't think your perspective is radical enough from a truly philosophical perspective. You take too much for granted and embrace too many idees recues. Thank you! Best! Jairo Jose da Silva

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Dear Jairo Jose de Silva,

The main contribution in the essay does not concern the application of numbers to physical states, but rather the application of geometrical descriptions. I used enumeration as an example, but it is not the main focus. So the issue of a "unit determination" never arises. Perhaps you do not find the ideas radical enough—one can try to be more radical—but the notion that space-time has an intrinsic geometrical structure is coherent and consistent. Given the right geometrical concepts, one can also see how temporal structure can generate such a geometry.

The problem with "transcendental" arguments, at least as Kant deployed them, is that they were supposed to explain how we can have various sorts of a priori knowledge. But as it turns out, we just don't have that knowledge. So the transcendental approach does not fit with what we now know.

Regards,

Tim Maudlin

The main contribution in the essay does not concern the application of numbers to physical states, but rather the application of geometrical descriptions. I used enumeration as an example, but it is not the main focus. So the issue of a "unit determination" never arises. Perhaps you do not find the ideas radical enough—one can try to be more radical—but the notion that space-time has an intrinsic geometrical structure is coherent and consistent. Given the right geometrical concepts, one can also see how temporal structure can generate such a geometry.

The problem with "transcendental" arguments, at least as Kant deployed them, is that they were supposed to explain how we can have various sorts of a priori knowledge. But as it turns out, we just don't have that knowledge. So the transcendental approach does not fit with what we now know.

Regards,

Tim Maudlin

Tim,

You say that "If there are physical items so constituted as to be solid objects, held together by strong internal forces and resistant to fracture and to amalgamation, then they will be effectively countable"; and that "The relevance of the theory of integers for physics is unproblematic so long as the way that physical items are being counted is conceptually sharp."

But counting implies making distinctions (1) and counting is necessarily a multi-step procedure, - so who or what is performing the counting procedure?

Lorraine Ford

1. "What is a Number?" by Louis H. Kauffman, http://homepages.math.uic.edu/~kauffman/NUM.html

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You say that "If there are physical items so constituted as to be solid objects, held together by strong internal forces and resistant to fracture and to amalgamation, then they will be effectively countable"; and that "The relevance of the theory of integers for physics is unproblematic so long as the way that physical items are being counted is conceptually sharp."

But counting implies making distinctions (1) and counting is necessarily a multi-step procedure, - so who or what is performing the counting procedure?

Lorraine Ford

1. "What is a Number?" by Louis H. Kauffman, http://homepages.math.uic.edu/~kauffman/NUM.html

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Dear Lorriane,

I would not say that a sharply defined enumeration requires any agent or anyone performing anything at all. What is required is the existence of a well-defined map from a physical situation to the integers, and the question is what physical characteristics the situation must have for the map to be well-defined. If I put a bunch of jelly beans in a jar and say there is a definite, exact number of jelly beans in the jar, that is true whether or not anyone ever counts them or goes through any multi-step procedure. It is because there already exists a definite number of jelly beans in the jar that if we want to find out how many there are, it does not matter who does the counting as long as they count correctly, or how they do it. The number they come up with will be the number that is already there before they start counting. If the jelly beans start to melt and merge then (again quite apart from anyone doing anything) the conditions required for a definite number may no longer obtain.

Regards,

Tim

I would not say that a sharply defined enumeration requires any agent or anyone performing anything at all. What is required is the existence of a well-defined map from a physical situation to the integers, and the question is what physical characteristics the situation must have for the map to be well-defined. If I put a bunch of jelly beans in a jar and say there is a definite, exact number of jelly beans in the jar, that is true whether or not anyone ever counts them or goes through any multi-step procedure. It is because there already exists a definite number of jelly beans in the jar that if we want to find out how many there are, it does not matter who does the counting as long as they count correctly, or how they do it. The number they come up with will be the number that is already there before they start counting. If the jelly beans start to melt and merge then (again quite apart from anyone doing anything) the conditions required for a definite number may no longer obtain.

Regards,

Tim

Hi Tim,

Re "there already exists a definite number of jelly beans in the jar":

Where or how does this number exist? Seemingly this number also has a category: "jelly beans in the jar". How is this 2-part (i.e. number and category) informational entity that exists interconnected to physical reality?

Cheers,

Lorraine

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Re "there already exists a definite number of jelly beans in the jar":

Where or how does this number exist? Seemingly this number also has a category: "jelly beans in the jar". How is this 2-part (i.e. number and category) informational entity that exists interconnected to physical reality?

Cheers,

Lorraine

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HI Lorraine,

I'm not sure how to take the question "how and when does this number exist?". There is a particular physical situation—jelly beans in a jar—and it is clear where and how that physical situation came to exist. It came into existence when the jelly beans were put in the jar by a certain physical act. The point is that the physical situation so created is, because of its physical character, one in which a particular number can be used to describe the situation. The number itself, the thing being used to as part of a representation of the physical state, is not the sort of the thing that exists anywhere or is created. It is, if you will, a Platonic entity. Wigner does not doubt the existence of such Platonic mathematical entities, he just wonders why some of them would be of any use in describing physical situations. The "interconnection" here is the connection between a representation and a represented object. The representation is mathematical and the object physical. It is not a physical connection between the number and the physical situation, if that is what you have in mind, but a representational connection.

Framed this way, the only remaining questions is why certain physical situations or entities have a structure that is usefully represented using particular mathematical entities. That is the question I am trying to clarify, particularly for geometrical structure.

Regards,

Tim

I'm not sure how to take the question "how and when does this number exist?". There is a particular physical situation—jelly beans in a jar—and it is clear where and how that physical situation came to exist. It came into existence when the jelly beans were put in the jar by a certain physical act. The point is that the physical situation so created is, because of its physical character, one in which a particular number can be used to describe the situation. The number itself, the thing being used to as part of a representation of the physical state, is not the sort of the thing that exists anywhere or is created. It is, if you will, a Platonic entity. Wigner does not doubt the existence of such Platonic mathematical entities, he just wonders why some of them would be of any use in describing physical situations. The "interconnection" here is the connection between a representation and a represented object. The representation is mathematical and the object physical. It is not a physical connection between the number and the physical situation, if that is what you have in mind, but a representational connection.

Framed this way, the only remaining questions is why certain physical situations or entities have a structure that is usefully represented using particular mathematical entities. That is the question I am trying to clarify, particularly for geometrical structure.

Regards,

Tim

Dear Dr. Maudlin,

Perhaps you would be willing to say more about the relationship between the geometry of linear structures and the question whether time is asymmetric. I am concerned about the position, presented by Huw Price and others, that there is no intrinsic difference between the directions of past to future and of future to past. It seems to me that a geometry based on the concept of the line might be more helpful on this issue than a geometry based on the concept of the open set. In the first place, a linear-structure geometry could clarify the discussion. We could formulate one very basic issue as the question whether the mathematical structures more accurately corresponding to physical time are directed lines or undirected lines. Furthermore, as you point out, the theory of linear structures would be appropriate for either answer. So, just as with the question of continuity or discreteness, the mathematical language would not add its own weight to the scales when we are trying to investigate a problem about physical facts. As you said in response to an earlier comment, this neutrality is an advantage.

Best wishes,

Laurence Hitterdale

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Perhaps you would be willing to say more about the relationship between the geometry of linear structures and the question whether time is asymmetric. I am concerned about the position, presented by Huw Price and others, that there is no intrinsic difference between the directions of past to future and of future to past. It seems to me that a geometry based on the concept of the line might be more helpful on this issue than a geometry based on the concept of the open set. In the first place, a linear-structure geometry could clarify the discussion. We could formulate one very basic issue as the question whether the mathematical structures more accurately corresponding to physical time are directed lines or undirected lines. Furthermore, as you point out, the theory of linear structures would be appropriate for either answer. So, just as with the question of continuity or discreteness, the mathematical language would not add its own weight to the scales when we are trying to investigate a problem about physical facts. As you said in response to an earlier comment, this neutrality is an advantage.

Best wishes,

Laurence Hitterdale

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Dear Laurence,

As I see it, the situation is this. There are three possible positions on the direction of time: 1) there is no directionality to time at all—the direction from this event to the past is physically just like the direction from this event to the future—; 2) there is a directionality, but it is not fundamental but rather should be analyzed in terms of something else (e.g....

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As I see it, the situation is this. There are three possible positions on the direction of time: 1) there is no directionality to time at all—the direction from this event to the past is physically just like the direction from this event to the future—; 2) there is a directionality, but it is not fundamental but rather should be analyzed in terms of something else (e.g....

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Dear Dr. Maudlin,

Thank you. Your response is helpful. I tend to favor either the second or third possible position, probably inclining more towards the second of the three. I realize that the availability of a particular mathematical structure is not of itself an argument that an aspect of physical reality is one way rather than another. Still, it is important, I think, that we are not compelled to use a mathematical language which, if time is inherently asymmetrical, leaves that asymmetry out of the mathematical representation. It is good to know that there is an alternative language which would capture this important feature of the physical reality.

Best wishes,

Laurence Hitterdale

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Thank you. Your response is helpful. I tend to favor either the second or third possible position, probably inclining more towards the second of the three. I realize that the availability of a particular mathematical structure is not of itself an argument that an aspect of physical reality is one way rather than another. Still, it is important, I think, that we are not compelled to use a mathematical language which, if time is inherently asymmetrical, leaves that asymmetry out of the mathematical representation. It is good to know that there is an alternative language which would capture this important feature of the physical reality.

Best wishes,

Laurence Hitterdale

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Dear Tim Maudlin,

When David Joyce commented on a previous essay of mine "it contains some interesting points", I was not sure whether at least he understood my observation that Dedekind replaced Euclid's 1-D notion of number as a measure (or as you are calling it a line?) by the 0-D point at the end of the distance from zero. I understand that it might be no opportune to question the fundamental of point set theory and point set topology. Did you deal with this perhaps historically decisive change?

Since I read Fraenkel 1923, I am sure to understand Cantor's logical flaw. My strongest additional argument is the indisputable fact that alephs in excess of 1 didn't prove useful.

Concerning my distinction between Relativity and relativity, see the essay by Phipps. My opinion that there is no preferred point in space but the natural zero of elapsed as well as future time might be too bewildering to those like you.

Because English is not my mother tongue, I had sometimes difficulties to clearly understand what you meant, e.g. on p. 5 with "sifting humor". On the same page, it would be helpful to find out where footnote 2 refers to and what conjugate points are meant.

Just an aside concerning Wigner: Von Békésy got a Nobel prize for his claim of a a passive traveling wave in cochlea, the mathematics of which was provided by Lighthill and was indeed unreasonably effective in the sense it was just fitted to measured data. Already Thomas Gold had argued that a passive traveling wave cannot work at all. Later on the cochlear amplifier was found.

Respectfully,

Eckard Blumschein

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When David Joyce commented on a previous essay of mine "it contains some interesting points", I was not sure whether at least he understood my observation that Dedekind replaced Euclid's 1-D notion of number as a measure (or as you are calling it a line?) by the 0-D point at the end of the distance from zero. I understand that it might be no opportune to question the fundamental of point set theory and point set topology. Did you deal with this perhaps historically decisive change?

Since I read Fraenkel 1923, I am sure to understand Cantor's logical flaw. My strongest additional argument is the indisputable fact that alephs in excess of 1 didn't prove useful.

Concerning my distinction between Relativity and relativity, see the essay by Phipps. My opinion that there is no preferred point in space but the natural zero of elapsed as well as future time might be too bewildering to those like you.

Because English is not my mother tongue, I had sometimes difficulties to clearly understand what you meant, e.g. on p. 5 with "sifting humor". On the same page, it would be helpful to find out where footnote 2 refers to and what conjugate points are meant.

Just an aside concerning Wigner: Von Békésy got a Nobel prize for his claim of a a passive traveling wave in cochlea, the mathematics of which was provided by Lighthill and was indeed unreasonably effective in the sense it was just fitted to measured data. Already Thomas Gold had argued that a passive traveling wave cannot work at all. Later on the cochlear amplifier was found.

Respectfully,

Eckard Blumschein

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Dear Eckart Blumenschein,

I can reposed quickly to the questions about my essay. The term "sifting humor" was used by David Hume, and it means continuing to analyze some concepts even further. Readers familiar with Hume would pick that up, but probably if you had not read the passage in Hume it would sound odd even to a native speaker.

The conjugate points I have in mind can occur in models of General Relativity (but not Special Relativity) where distinct light-like geodesics intersect more than once. Call two such intersection points A and B. In such a case, each path is light-like even though A and B are the endpoints of two different lines (in my sense). So the criterion for a light-like geodesic that works well in Special Relativity fails in General Relativity. But this can be fixed, because each light-like geodesic can be subdivided into overlapping parts, each of which satisfies the simple definition. So the General Relativistic case can be covered by a simple amendment to the definition.

Regards,

Tim Maudlin

I can reposed quickly to the questions about my essay. The term "sifting humor" was used by David Hume, and it means continuing to analyze some concepts even further. Readers familiar with Hume would pick that up, but probably if you had not read the passage in Hume it would sound odd even to a native speaker.

The conjugate points I have in mind can occur in models of General Relativity (but not Special Relativity) where distinct light-like geodesics intersect more than once. Call two such intersection points A and B. In such a case, each path is light-like even though A and B are the endpoints of two different lines (in my sense). So the criterion for a light-like geodesic that works well in Special Relativity fails in General Relativity. But this can be fixed, because each light-like geodesic can be subdivided into overlapping parts, each of which satisfies the simple definition. So the General Relativistic case can be covered by a simple amendment to the definition.

Regards,

Tim Maudlin

Dear Tim Maudlin,

Thank you for your quick response. Lee Smolin lost my admiration because up to now he did not even respond to my simple request whether he actually meant "off", what was not understandable to me, or simply "of". I wonder why nobody else admitted not having understood your term "sifting humor".

We all make mistakes. Misspellings of my name don't matter; here is no risk of confusion.

I am looking forward learning from your criticism of my admittedly uncommon arguments.

Regards,

Eckard Blumschein

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Thank you for your quick response. Lee Smolin lost my admiration because up to now he did not even respond to my simple request whether he actually meant "off", what was not understandable to me, or simply "of". I wonder why nobody else admitted not having understood your term "sifting humor".

We all make mistakes. Misspellings of my name don't matter; here is no risk of confusion.

I am looking forward learning from your criticism of my admittedly uncommon arguments.

Regards,

Eckard Blumschein

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Dear Tim and dear Spencer,

Are mathematicians in position to at least partially correct mistakes? I was surprised reading in Wikipedia: "once known as the topology of point sets, this usage is now obsolete".

Of course, while it is possible to attribute a direction to a measure e.g. to a piece of a line, a point doesn't have a direction. Moreover, a set of continuous 1-D pieces can constitute any continuous line while Dedekind just begged to believe that a "dense" amount of 0-D points may constitute a continuous line.

I agree with Spencer on that the symmetry between past and future cannot be avoided just by means of the otherwise necessary return to Euclid's notion of number. I tried to explain in a discussion with Cristinel Stoica that the direction got inevitably lost due to abstraction from reality to model.

Serious criticism of my arguments is welcome.

Regards,

Eckard

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Are mathematicians in position to at least partially correct mistakes? I was surprised reading in Wikipedia: "once known as the topology of point sets, this usage is now obsolete".

Of course, while it is possible to attribute a direction to a measure e.g. to a piece of a line, a point doesn't have a direction. Moreover, a set of continuous 1-D pieces can constitute any continuous line while Dedekind just begged to believe that a "dense" amount of 0-D points may constitute a continuous line.

I agree with Spencer on that the symmetry between past and future cannot be avoided just by means of the otherwise necessary return to Euclid's notion of number. I tried to explain in a discussion with Cristinel Stoica that the direction got inevitably lost due to abstraction from reality to model.

Serious criticism of my arguments is welcome.

Regards,

Eckard

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Dear Tim:

This is a fantastic essay. And it is very well written.

As we all know, mathematics has been very effective in physics. Its weaknesses to date in modelling physical reality have been twofold:

1. Using open set theory, it does not model that time has an order (whether we interpret this as an order in forward time or an order in backward time);

2. It does not model which directed order (forward order or backward order) corresponds to the observed Arrow of Time.

Your Theory of Linear Structures addresses point 1 - and is therefore important. However, I do not believe it can address the second point. In particular, the initial end point of a line could represent a past instant of time and the final end point of the line could represent a future instant of time - but equally the initial end point of a line could represent a future instant of time and the final end point of the line could represent a past instant of time. The mathematics cannot differentiate between these two interpretations. So, the Theory of Linear Structures, while it can encode an order, will not I believe be able to encode which directed order of a line corresponds to the observed Arrow of Time. Instead, we would have to impose the direction of time from outside the mathematics on the solution - as we do now for open set theory.

What this would mean is that the Theory of Linear Structures - although important - will produce time-symmetric theories, as open set theory does now. We will unfortunately not be able to use it, for example, to have a mathematical theory of evolution. Nevertheless, your work is very good.

If you are interested, in my essay I explain more generally why mathematics cannot, in principle, model the Arrow of Time.

Thank you again for some great research and a clear essay.

Kind regards

Spencer Scoular

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This is a fantastic essay. And it is very well written.

As we all know, mathematics has been very effective in physics. Its weaknesses to date in modelling physical reality have been twofold:

1. Using open set theory, it does not model that time has an order (whether we interpret this as an order in forward time or an order in backward time);

2. It does not model which directed order (forward order or backward order) corresponds to the observed Arrow of Time.

Your Theory of Linear Structures addresses point 1 - and is therefore important. However, I do not believe it can address the second point. In particular, the initial end point of a line could represent a past instant of time and the final end point of the line could represent a future instant of time - but equally the initial end point of a line could represent a future instant of time and the final end point of the line could represent a past instant of time. The mathematics cannot differentiate between these two interpretations. So, the Theory of Linear Structures, while it can encode an order, will not I believe be able to encode which directed order of a line corresponds to the observed Arrow of Time. Instead, we would have to impose the direction of time from outside the mathematics on the solution - as we do now for open set theory.

What this would mean is that the Theory of Linear Structures - although important - will produce time-symmetric theories, as open set theory does now. We will unfortunately not be able to use it, for example, to have a mathematical theory of evolution. Nevertheless, your work is very good.

If you are interested, in my essay I explain more generally why mathematics cannot, in principle, model the Arrow of Time.

Thank you again for some great research and a clear essay.

Kind regards

Spencer Scoular

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Dear Spencer,

Thanks for the careful reading. I don't think we disagree on this much: there is a real, physical distinction between the two time directions and it would be useful to have a way to describe geometries that include such a fundamental asymmetry. That is the part my language does. Then we ask a further question: can we give a more profound account of the physical nature of the asymmetry? This could turn out two ways: it could be a fundamental physical structure, and so not admit of further analysis, or there may be a deeper analysis possible. I am open to both possibilities, and it sounds like your work addresses the second.

My hope is that this mathematical structure can be of use for many different projects. Perhaps yours is one.

Cheers,

Tim

Thanks for the careful reading. I don't think we disagree on this much: there is a real, physical distinction between the two time directions and it would be useful to have a way to describe geometries that include such a fundamental asymmetry. That is the part my language does. Then we ask a further question: can we give a more profound account of the physical nature of the asymmetry? This could turn out two ways: it could be a fundamental physical structure, and so not admit of further analysis, or there may be a deeper analysis possible. I am open to both possibilities, and it sounds like your work addresses the second.

My hope is that this mathematical structure can be of use for many different projects. Perhaps yours is one.

Cheers,

Tim

Dear Tim Maudlin,

I know from your work that you have a strong acquaintance withh Bell's work (B). I arrived at Bell/CHSH inequality from my investigation of Kochen-Specker theorem for multiple qubits mainly through Mermin' treatise (my ref. [19]). At some stage, I observed that the commutation diagram for a set of four observables involved in the violation of the inequality is just a...

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I know from your work that you have a strong acquaintance withh Bell's work (B). I arrived at Bell/CHSH inequality from my investigation of Kochen-Specker theorem for multiple qubits mainly through Mermin' treatise (my ref. [19]). At some stage, I observed that the commutation diagram for a set of four observables involved in the violation of the inequality is just a...

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Dear Michel,

Maybe it is because you came at this from KS that this seems unfamiliar. If one thinks of the sort of experimental arrangement that Bell had in mind, with observation on the two qubits being made very far apart, then the commutation structure you mention is obvious: any observable on one side must commute with any observable on the other, or else qm would violate no-signalling. And on the same side, the two possible observables cannot commute, or else you do not violate the inequality. But it is not the case that for any set of observables with the commutation structure you show that one can get the maximal qm violation of 2tr(2). so the norm you mention does not follow from the commutation structure you have written down. (Think or what happens as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2, not 2rt(2).)

Regards,

Tim

Maybe it is because you came at this from KS that this seems unfamiliar. If one thinks of the sort of experimental arrangement that Bell had in mind, with observation on the two qubits being made very far apart, then the commutation structure you mention is obvious: any observable on one side must commute with any observable on the other, or else qm would violate no-signalling. And on the same side, the two possible observables cannot commute, or else you do not violate the inequality. But it is not the case that for any set of observables with the commutation structure you show that one can get the maximal qm violation of 2tr(2). so the norm you mention does not follow from the commutation structure you have written down. (Think or what happens as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2, not 2rt(2).)

Regards,

Tim

Dear Tim,

I agree with your point (i): commutation, in my diagram IX commute with XI and ZI but not with IZ, and vice versa.

I don't agree with point (ii) for qubits. I have checked that for all multiple qubit operators (starting with two qubits) one arrives at the maximal violation 2v2. It is the reason why a finite geometry like the Mermin's square (the 3 by 3 grid) for two qubits has nine proofs of Bell's theorem in it.

If one makes use of the dessins d'enfants the extension field involved is Q(v2).

You may have in mind another experimental scheme than the one I am using where the maximum violation does not apply stricto sensu.

For other type of violations of classical inequalities, there is the paper by Alexis Grunbaum and an optical experiment that I mention in his blog.

Michel

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I agree with your point (i): commutation, in my diagram IX commute with XI and ZI but not with IZ, and vice versa.

I don't agree with point (ii) for qubits. I have checked that for all multiple qubit operators (starting with two qubits) one arrives at the maximal violation 2v2. It is the reason why a finite geometry like the Mermin's square (the 3 by 3 grid) for two qubits has nine proofs of Bell's theorem in it.

If one makes use of the dessins d'enfants the extension field involved is Q(v2).

You may have in mind another experimental scheme than the one I am using where the maximum violation does not apply stricto sensu.

For other type of violations of classical inequalities, there is the paper by Alexis Grunbaum and an optical experiment that I mention in his blog.

Michel

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Dear Michel,

Consider the following experimental set-up. On one side, there is a choice between measuring spin in the z-direction and in a direction 5° away from the z-direction, and similarly on the other side. Since the two possible measurement on each side do not commute, and each on one side commute with both on the other, this satisfies your commutation square. But no quantum state gives the maximal value of 2rt(2). If you think one does, maybe you can specify what you think it is.

Cheers,

Tim

Consider the following experimental set-up. On one side, there is a choice between measuring spin in the z-direction and in a direction 5° away from the z-direction, and similarly on the other side. Since the two possible measurement on each side do not commute, and each on one side commute with both on the other, this satisfies your commutation square. But no quantum state gives the maximal value of 2rt(2). If you think one does, maybe you can specify what you think it is.

Cheers,

Tim

Dear Tim,

More on your comment: "as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2": this is classical argument that seems irrelevant in the quantum (not spatial) scheme, either the spins are apart or the same.

Best,

Michel

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More on your comment: "as the two possible spin directions on each side approach each other: in the limit, the maximal value becomes 2": this is classical argument that seems irrelevant in the quantum (not spatial) scheme, either the spins are apart or the same.

Best,

Michel

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Dear Tim,

You write a very attractive essay about a speculated structure of the "physical space-time" that you call a Theory of (Directed) Linear Structures (DLS). " I will argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to space-time". Your essay is based on your recent book at Oxford University Press that unfortunately I could not access...

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You write a very attractive essay about a speculated structure of the "physical space-time" that you call a Theory of (Directed) Linear Structures (DLS). " I will argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to space-time". Your essay is based on your recent book at Oxford University Press that unfortunately I could not access...

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Dear Michel,

In the finite point case, the theory essentially reduces to directed graph theory. I don't see any deep connection to group theory and representations of cosets...the nearer analogs are differential geometry and (a bit) even non-commutative geometry (only the latter with strong restrictions). If you take as a target the space-times that are solutions to the General Relativistic Field equations, then there is not so much reason to focus on automorphisms.

The continuous case will include all standard Riemannian and semi-Riemannian structures. I can't see any connection to Cantor sets. I think that trying to connect this approach to group theory is not the right way to go.

Regards,

Tim

In the finite point case, the theory essentially reduces to directed graph theory. I don't see any deep connection to group theory and representations of cosets...the nearer analogs are differential geometry and (a bit) even non-commutative geometry (only the latter with strong restrictions). If you take as a target the space-times that are solutions to the General Relativistic Field equations, then there is not so much reason to focus on automorphisms.

The continuous case will include all standard Riemannian and semi-Riemannian structures. I can't see any connection to Cantor sets. I think that trying to connect this approach to group theory is not the right way to go.

Regards,

Tim

Hello Tim,

I found a lot to like, in your essay. You articulated well, the problems I've encountered with point set topology, and its limitations for a realistic description of physical form. I have adopted a constructivist and emergentist view toward geometry, in my own research, which reproduces some of the features of your linear structure theory program. And intriguingly; my work linking Cosmology to the Mandelbrot Set necessitated a departure from the standard program, and conclusions similar to yours.

The Mandelbrot Mapping Conjecture states that the periphery of the Mandelbrot Set encodes the dynamism for the full range of physical processes, from the most to the least energetic. So if it is rotated from the conventional view such that (-2,0i) is on the bottom; it can be viewed as a thermometer. But in cosmological terms; the cusp at (.25,0i) is the moment of the universe's inception. What is clearly described, even when the argument is extended into the quaternion and octonion domains, is that initially spacetime was purely timelike and broken symmetry forced spacetime to evolve relativistically.

Philosophically speaking; we know that for structures to exist in time, they must have a time-like projection or duration. Accordingly; for spacetime itself to be an enduring feature, it must also exist in time and have a time-like aspect - hence it must possess linear structure. This is something I have attempted to articulate in several papers, but you have summed things up rather elegantly. I will have to take some time to re-read this paper, and fully digest it, before I make a determination. I have some issues to address, I think. But on first look it appears your program has much to recommend it.

All the Best,

Jonathan

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I found a lot to like, in your essay. You articulated well, the problems I've encountered with point set topology, and its limitations for a realistic description of physical form. I have adopted a constructivist and emergentist view toward geometry, in my own research, which reproduces some of the features of your linear structure theory program. And intriguingly; my work linking Cosmology to the Mandelbrot Set necessitated a departure from the standard program, and conclusions similar to yours.

The Mandelbrot Mapping Conjecture states that the periphery of the Mandelbrot Set encodes the dynamism for the full range of physical processes, from the most to the least energetic. So if it is rotated from the conventional view such that (-2,0i) is on the bottom; it can be viewed as a thermometer. But in cosmological terms; the cusp at (.25,0i) is the moment of the universe's inception. What is clearly described, even when the argument is extended into the quaternion and octonion domains, is that initially spacetime was purely timelike and broken symmetry forced spacetime to evolve relativistically.

Philosophically speaking; we know that for structures to exist in time, they must have a time-like projection or duration. Accordingly; for spacetime itself to be an enduring feature, it must also exist in time and have a time-like aspect - hence it must possess linear structure. This is something I have attempted to articulate in several papers, but you have summed things up rather elegantly. I will have to take some time to re-read this paper, and fully digest it, before I make a determination. I have some issues to address, I think. But on first look it appears your program has much to recommend it.

All the Best,

Jonathan

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Dear Jonathan,

Thanks for the comments, and I hope you find something useful in the program. I would be a bit surprised if it makes contact with what you are doing given that you start from fractals. In fact, my approach tends to make fractals more peripheral than the standard topological approaches, because by the standard topological definition of "continuous" fractal functions are continuous and by my definition they are not. I have a little hope that this might help for a path-integral formulation of the quantum theory, because in the standard approach the measure over path space tends to be dominated by fractals, with makes it something of a mess. But if this is any use for your approach, I would be very pleased,

Regards.

Tim

Thanks for the comments, and I hope you find something useful in the program. I would be a bit surprised if it makes contact with what you are doing given that you start from fractals. In fact, my approach tends to make fractals more peripheral than the standard topological approaches, because by the standard topological definition of "continuous" fractal functions are continuous and by my definition they are not. I have a little hope that this might help for a path-integral formulation of the quantum theory, because in the standard approach the measure over path space tends to be dominated by fractals, with makes it something of a mess. But if this is any use for your approach, I would be very pleased,

Regards.

Tim

Bruce Lee once said..

Retain what is useful. I like that advice.

Regards,

Jonathan

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Retain what is useful. I like that advice.

Regards,

Jonathan

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Mountains may be hard to define precisely..

But observers on adjoining peaks can clearly distinguish their positions from each other, and have a distance to travel to be in the same place. I find that pondering questions like 'what is the distance when traveling along the shore in Britain?' make life interesting for me.

All the Best,

Jonathan

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But observers on adjoining peaks can clearly distinguish their positions from each other, and have a distance to travel to be in the same place. I find that pondering questions like 'what is the distance when traveling along the shore in Britain?' make life interesting for me.

All the Best,

Jonathan

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Dear Tim,

The number of topologies (transitive digraphs) on n labeled elements is Sloane sequence 1000798 that is 1, 1, 4, 29, 355, 6942, 209527, ... for n = 1, 2, 3, 4, 5, 6, 7. The number of DLS for n =5 is 2^20 = 1,048,576.

Do you always have the number of DLS equals to 2^p (for some p) and what is the sequence? I suspect a relation to a finite projective space PG(2,p-1), e.g. for n = 5 the number of DLS is 1+|PG(2,19)|. This is reminiscent of a Veldkamp space (set of hyperplanes of a finite geometry with 3 points on a line).

Regards,

Michel

ps/ Today an interesting paper about topologies on a finite set

http://xxx.lanl.gov/pdf/1503.08359.pdf

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The number of topologies (transitive digraphs) on n labeled elements is Sloane sequence 1000798 that is 1, 1, 4, 29, 355, 6942, 209527, ... for n = 1, 2, 3, 4, 5, 6, 7. The number of DLS for n =5 is 2^20 = 1,048,576.

Do you always have the number of DLS equals to 2^p (for some p) and what is the sequence? I suspect a relation to a finite projective space PG(2,p-1), e.g. for n = 5 the number of DLS is 1+|PG(2,19)|. This is reminiscent of a Veldkamp space (set of hyperplanes of a finite geometry with 3 points on a line).

Regards,

Michel

ps/ Today an interesting paper about topologies on a finite set

http://xxx.lanl.gov/pdf/1503.08359.pdf

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Dear Tim,

I found your slides of a 2013 talk. The number of DLS is 1,4,64,4096,1048576 (Sloane sequence 1053763). It is the number of simple digraphs (without self-loops) on n labeled nodes. It also corresponds to the number of nilpotent n x n matrices over GF(2).

May be this property of nilpotency makes sense for a space time as it does for quantum mechanics (Rowlands). Your approach opens many perspectives.

Michel

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I found your slides of a 2013 talk. The number of DLS is 1,4,64,4096,1048576 (Sloane sequence 1053763). It is the number of simple digraphs (without self-loops) on n labeled nodes. It also corresponds to the number of nilpotent n x n matrices over GF(2).

May be this property of nilpotency makes sense for a space time as it does for quantum mechanics (Rowlands). Your approach opens many perspectives.

Michel

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Hi Tim--

Your essay is superb: rigorous yet readable. In particular, I thought that it was quite thought-provoking, which is the sure sign of any excellent essay.

Question: How does your theory of Theory of Linear Structures deal with closed time-like curves (CTCs) in General Relativity? To focus the question, consider the Gödel Cosmology, which you addressed, for example, on pp. 216-217 of Quantum Non-Locality & Relativity (3rd Ed.). As I understand it, Gödel formulated his cosmology to put a stake through the heart of time, specifically, the notion that time consists of well-ordered linear events. This seems to directly conflict with your theory. Of course, there have been many responses to Gödel’s Universe. Many a physicist has simply noted that the Universe does not appear to rotate. Others have said that the cosmology is so insanely vicious that it just can’t be right. What’s your view?

Two additional, non-substantive points:

First, don’t forget page numbers!

Second, on a very personal note, I would like to thank you for taking the time to respond to everyone’s questions. You are one of the most prominent contestants. (I know that because I own several of your books, including both the second and third editions of QNL&R.) We all know that you are a very busy guy. Nonetheless, you have taken the time to patiently and judiciously respond to all manner of posts. That is incredibly impressive.

Very best regards,

Bill.

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Your essay is superb: rigorous yet readable. In particular, I thought that it was quite thought-provoking, which is the sure sign of any excellent essay.

Question: How does your theory of Theory of Linear Structures deal with closed time-like curves (CTCs) in General Relativity? To focus the question, consider the Gödel Cosmology, which you addressed, for example, on pp. 216-217 of Quantum Non-Locality & Relativity (3rd Ed.). As I understand it, Gödel formulated his cosmology to put a stake through the heart of time, specifically, the notion that time consists of well-ordered linear events. This seems to directly conflict with your theory. Of course, there have been many responses to Gödel’s Universe. Many a physicist has simply noted that the Universe does not appear to rotate. Others have said that the cosmology is so insanely vicious that it just can’t be right. What’s your view?

Two additional, non-substantive points:

First, don’t forget page numbers!

Second, on a very personal note, I would like to thank you for taking the time to respond to everyone’s questions. You are one of the most prominent contestants. (I know that because I own several of your books, including both the second and third editions of QNL&R.) We all know that you are a very busy guy. Nonetheless, you have taken the time to patiently and judiciously respond to all manner of posts. That is incredibly impressive.

Very best regards,

Bill.

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Dear Bill,

Thanks for the kinds words!

The situation with respect to CTCs is interesting, and it goes like this:

If all you want to do is model a space-time with CTCs, you can do it. In fact, there is clean definition of a "simple loop" in this theory: a simple loop is a set of points that is not itself a line, but removing any point from the set yields a line. (Recall that...

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Thanks for the kinds words!

The situation with respect to CTCs is interesting, and it goes like this:

If all you want to do is model a space-time with CTCs, you can do it. In fact, there is clean definition of a "simple loop" in this theory: a simple loop is a set of points that is not itself a line, but removing any point from the set yields a line. (Recall that...

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Dear Bill,

In case it wasn't clear: the point about simple loops is that in this approach the existence of a CTC is just the existence of a simple loop in the geometry. We get rid of all space-like lines and space-like geometry, and leave only lines that are everywhere time-like or null. So we can model such simple loops, but admitting them spoils a nice program for deriving the whole geometry from temporal structure.

Cheers,

Tim

In case it wasn't clear: the point about simple loops is that in this approach the existence of a CTC is just the existence of a simple loop in the geometry. We get rid of all space-like lines and space-like geometry, and leave only lines that are everywhere time-like or null. So we can model such simple loops, but admitting them spoils a nice program for deriving the whole geometry from temporal structure.

Cheers,

Tim

Dear Sir, When I read the first sentence of the abstract I am certain that your thesis is a tautology. Mathematics is a human invention, and as such is fallible. Your thesis doesn't seem to take account of that problem. In my view, any attempt to expound the thesis of the essay is bound to be problematic, because the proposed subject implies that mathematics is physics, or put differently that the universe is mathematical. Since all of the ideas involved are human inventions, they are likely to be completely wrong in conception. So far I have found nothing to convince me that the inventions that we humans have created do represent actual truth and so are not just fallible delusions of the human imagination.

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"Our understanding of the structure of time has been revolutionized by the Theory of Relativity. Intriguingly, the change from a classical to a Relativistic account of temporal structure is of exactly the right sort to promote time into the sole creator of physical geometry."

How did the theory of relativity "revolutionize" the understanding of time? By replacing the true tenet of Newton's...

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How did the theory of relativity "revolutionize" the understanding of time? By replacing the true tenet of Newton's...

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Dear Pentcho Valev.

We know that the speed of light is not affected by the speed of the emitter by the observation (for example) of binary stars. If the speed of light emitted by the receding star were even slightly different from that of the approaching star, given the period of time that the light is en route, the apparent motions of the stars as seen from earth would be quite different from what we see. The motions would not appear to us to be regular. So the independence of the trajectory of a light ray in a vacuum from the motion of the emitter is empirically established.

Regards,

Tim Maudlin

We know that the speed of light is not affected by the speed of the emitter by the observation (for example) of binary stars. If the speed of light emitted by the receding star were even slightly different from that of the approaching star, given the period of time that the light is en route, the apparent motions of the stars as seen from earth would be quite different from what we see. The motions would not appear to us to be regular. So the independence of the trajectory of a light ray in a vacuum from the motion of the emitter is empirically established.

Regards,

Tim Maudlin

"The de Sitter effect was described by de Sitter in 1913 and used to support the special theory of relativity against a competing 1908 emission theory by Walter Ritz that postulated a variable speed of light. De Sitter showed that Ritz's theory predicted that the orbits of binary stars would appear more eccentric than consistent with experiment and with the laws of mechanics. (...) De Sitter's argument was criticized because of possible extinction effects. That is, during their flight to Earth, the light rays should have been absorbed and re-emitted by interstellar matter nearly at rest relative to Earth, so that the speed of light should become constant with respect to Earth. However, Kenneth Brecher published the results of a similar double-survey in 1977, and reached a similar conclusion - that any apparent irregularities in double-star orbits were too small to support the emission theory. Contrary to De Sitter, he observed the x-ray spectrum, thereby eliminating possible influences of the extinction effect."

Here is Brecher's paper:

K. Brecher, "Is the Speed of Light Independent of the Velocity of the Source?"

Brecher (originally de Sitter) expects a system with unknown parameters to produce "peculiar effects". The system does not produce them. Conclusion: Ritz's emission theory (more precisely, the assumption that the speed of light depends on the speed of the emitter) is unequivocally refuted, Einstein's theory is gloriously confirmed.

Needless to say, refutations and confirmations of this kind can only be valid in Einstein's world. Note that they cannot be criticized - the fact that the parameters of the double star system are unknown does not allow critics to show why exactly the "peculiar effects" are absent.

Pentcho Valev

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Here is Brecher's paper:

K. Brecher, "Is the Speed of Light Independent of the Velocity of the Source?"

Brecher (originally de Sitter) expects a system with unknown parameters to produce "peculiar effects". The system does not produce them. Conclusion: Ritz's emission theory (more precisely, the assumption that the speed of light depends on the speed of the emitter) is unequivocally refuted, Einstein's theory is gloriously confirmed.

Needless to say, refutations and confirmations of this kind can only be valid in Einstein's world. Note that they cannot be criticized - the fact that the parameters of the double star system are unknown does not allow critics to show why exactly the "peculiar effects" are absent.

Pentcho Valev

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Actually, the Relativistic treatment of binaries has been quite strongly tested, apart from just general considerations about how gravitating bodes orbit, including precise predictions for changes in orbital period due to gravitational waves. And of course, the gravitational temporal effects are now confirmed using high-accuracy clocks even just in the Earth's gravitational field.

I am open to emendations to the Relativistic picture—indeed, I think quantum non-locality suggests it—but the basic Relativistic account of temporal structure has been severely tested in many distinct ways, and seems to be close to correct.No alternative does so well.

I am open to emendations to the Relativistic picture—indeed, I think quantum non-locality suggests it—but the basic Relativistic account of temporal structure has been severely tested in many distinct ways, and seems to be close to correct.No alternative does so well.

Hi Tim,

My idea is also based on the concept of line, however the line in my system has a very simple interpretation, its the difference between two quantities that is all. The quantities have to be random otherwise you will not get our reality. Do you have some implementation of your system so I may compare it to mine.

I get real physics from my system.If you don't have the time just read the electron mass section and run the program (click "program link" at the end of the section) , it will execute in less than a minute.

Essay

Thanks and good luck.

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My idea is also based on the concept of line, however the line in my system has a very simple interpretation, its the difference between two quantities that is all. The quantities have to be random otherwise you will not get our reality. Do you have some implementation of your system so I may compare it to mine.

I get real physics from my system.If you don't have the time just read the electron mass section and run the program (click "program link" at the end of the section) , it will execute in less than a minute.

Essay

Thanks and good luck.

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Hi Adel,

Part of this project is to see how much of the geometrical structure of a space can be represented without there being any "quantities" at all, that is, it is done at a sub-metrical level. So if you are starting with quantities that have well-defined differences, your starting point is quite different from mine, and the approaches probably will not coincide. In fact, one of my goals was the opposite of Dedekind's. Dedekind wanted to get all reference to geometry out of his theory of numbers, and I want to eliminate all reference to numerical structure (including differences) from my account of geometry, or at least to have a very clear understanding of how any numerical structure gets in.

Cheers,

Tim

Part of this project is to see how much of the geometrical structure of a space can be represented without there being any "quantities" at all, that is, it is done at a sub-metrical level. So if you are starting with quantities that have well-defined differences, your starting point is quite different from mine, and the approaches probably will not coincide. In fact, one of my goals was the opposite of Dedekind's. Dedekind wanted to get all reference to geometry out of his theory of numbers, and I want to eliminate all reference to numerical structure (including differences) from my account of geometry, or at least to have a very clear understanding of how any numerical structure gets in.

Cheers,

Tim

Dear Tim, in the paragraph before last, you write:

"Wigner’s question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh."

I do not think that the way to answer Wigner's question lies on the plane expressed in this excerpt, which seems to reflect your approach. Mathematics suggests various rational structures. Physics tries them out for the role of laws of nature. Neither physics nor mathematics is suited to answer the 'why' question of Wigner: they both deal with 'how' problems only.

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"Wigner’s question is this: why is the language of mathematics so well suited to describe the physical world? A proper answer to this question must approach it from both directions: the direction of the mathematical language and the direction of the structure of the physical world being represented. In order for the language to fit the object in a useful way the two sides have to mesh."

I do not think that the way to answer Wigner's question lies on the plane expressed in this excerpt, which seems to reflect your approach. Mathematics suggests various rational structures. Physics tries them out for the role of laws of nature. Neither physics nor mathematics is suited to answer the 'why' question of Wigner: they both deal with 'how' problems only.

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Dear Alexey,

"Why" is used in lots of ways. If I ask "Why did the bridge collapse?" it typically means exactly the same as "How did it come about that the bridge collapsed" or "What are the conditions that account for the bridge's collapse?" and an answer such as "Because the bolts rusted through" is fine. This is the sense of "why" I had in mind. I think it is also what Wigner had in mind: recall that he had no problems about certain mathematical structures being useful for physics. So your distinction between "why" and "how" questions does not seem to me to be appropriate to Wigner's concerns.

Regards,

Tim

"Why" is used in lots of ways. If I ask "Why did the bridge collapse?" it typically means exactly the same as "How did it come about that the bridge collapsed" or "What are the conditions that account for the bridge's collapse?" and an answer such as "Because the bolts rusted through" is fine. This is the sense of "why" I had in mind. I think it is also what Wigner had in mind: recall that he had no problems about certain mathematical structures being useful for physics. So your distinction between "why" and "how" questions does not seem to me to be appropriate to Wigner's concerns.

Regards,

Tim

Well, Lo and Behold, as I was about to post my finished comment, I saw Tim’s answer. Since I already wrote it, I am posting it anyway…

Dear Alexey/Lev Burov,

Not that Tim needs any help in responding to this, but he is likely to take a different approach than the one I would like everyone to be aware of.

The issue I am addressing here is your claim that: “Neither physics...

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Dear Alexey/Lev Burov,

Not that Tim needs any help in responding to this, but he is likely to take a different approach than the one I would like everyone to be aware of.

The issue I am addressing here is your claim that: “Neither physics...

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In order to less speculatively answer Wigner's question I would like to remind of Otto de Guericke who argued for experiments instead of scholastics. Leifer's cut between useful and unwarranted speculative mathematics might become increasingly necessary.

In order to give an example I quote from Daniel Braun's essay:

"Set theory is part of mathematics, and applying set theory to itself is known to quickly run into trouble." Endquote.

Braun is nonetheless perhaps almost the only one who tried to ascribe physical meaning to aleph_2 and aleph_3.

Well, transfinite numbers evade experimental verification. Having critically read Fraenkel 1923 I localized Cantor's mistake and agree with Galileo Galilei on that infinity is a property that must not be used as a number: oo+1=oo.

Eckard

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In order to give an example I quote from Daniel Braun's essay:

"Set theory is part of mathematics, and applying set theory to itself is known to quickly run into trouble." Endquote.

Braun is nonetheless perhaps almost the only one who tried to ascribe physical meaning to aleph_2 and aleph_3.

Well, transfinite numbers evade experimental verification. Having critically read Fraenkel 1923 I localized Cantor's mistake and agree with Galileo Galilei on that infinity is a property that must not be used as a number: oo+1=oo.

Eckard

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Tim,

The physical world will not change at our command but is made malleable by changing the mathematical language used to formulate physics.Is the Theory of Linear Structures only one example of how this can be done? Time w/o the application of the theory creates the geometry of space-time? Is space a reaction to mass over time then?

I found theoretical physicist, Ilya Prigogine's quote from 2004: "I believe that what we do today depends on our image of the future, rather than the future depending on what we do today. We build our equations by our actions. These equations, and the future they represent, are not written in nature. In other words, time becomes construction." Is this relevant to your Theory?

It seems germane to me but I doubt that I have a full understanding.

Jim

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The physical world will not change at our command but is made malleable by changing the mathematical language used to formulate physics.Is the Theory of Linear Structures only one example of how this can be done? Time w/o the application of the theory creates the geometry of space-time? Is space a reaction to mass over time then?

I found theoretical physicist, Ilya Prigogine's quote from 2004: "I believe that what we do today depends on our image of the future, rather than the future depending on what we do today. We build our equations by our actions. These equations, and the future they represent, are not written in nature. In other words, time becomes construction." Is this relevant to your Theory?

It seems germane to me but I doubt that I have a full understanding.

Jim

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JIm,

I don't think that the mathematics we use to describe the physical world can change the world itself. The question is rather whether the mathematics we choose to use is well-suited to the structure of the world. Different mathematical structures fit with different physical structures, and we are seeking the right one. The Theory of Linear Structures provides a novel sort of language to describe geometry, and particularly the geometry of space-time.

Using this language, one can analyze the whole space-time structure (and so anything one would like to say about space) as determined by purely temporal structure. In this sense, "space" arises out of time, or more exactly out of the Linear Structure created by ordering events in time in a Relativistic way.

Prigogine's quote sounds like it concerns human action. Human action can, of course, determine many facts about the future: whether there is climate change of a certain kind, or war, etc. And how we act depends on our beliefs and desires about the future. But the basic laws of physics, and the fundamental physical constituents of the universe, are not within our power to change. The best we can hope for is to figure out what they are an describe them accurately.

Regards,

Tim

I don't think that the mathematics we use to describe the physical world can change the world itself. The question is rather whether the mathematics we choose to use is well-suited to the structure of the world. Different mathematical structures fit with different physical structures, and we are seeking the right one. The Theory of Linear Structures provides a novel sort of language to describe geometry, and particularly the geometry of space-time.

Using this language, one can analyze the whole space-time structure (and so anything one would like to say about space) as determined by purely temporal structure. In this sense, "space" arises out of time, or more exactly out of the Linear Structure created by ordering events in time in a Relativistic way.

Prigogine's quote sounds like it concerns human action. Human action can, of course, determine many facts about the future: whether there is climate change of a certain kind, or war, etc. And how we act depends on our beliefs and desires about the future. But the basic laws of physics, and the fundamental physical constituents of the universe, are not within our power to change. The best we can hope for is to figure out what they are an describe them accurately.

Regards,

Tim

Tim,

As time grows short, I revisit those I have read. I find that I did not rate yours, something I usually do to those that impress me, so I am rectifying that. Hope you get a chance to look at mine.

Jim

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As time grows short, I revisit those I have read. I find that I did not rate yours, something I usually do to those that impress me, so I am rectifying that. Hope you get a chance to look at mine.

Jim

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Dear Tim,

I enjoyed a lot your witty and refreshing essay. I too have mentioned the proof of Fermat's Last, but your most amusing and insightful conclusion about a companion to Wigner's paper didn't even cross my mind.

I think your treatment of time as fundamental is exciting and aiming to fill a gap in the foundations of today's physics. I know you have a book on the topic and I am planning to read it, however I am reasonably sure that the book, as any other topic-focused reading, does not answer a curiosity of mine regarding your view. I would like to know where is your intuition about time stemming from. I mean, when was that moment that your intuition crystallized and what caused it to happen? If you did write about it, I'd be grateful to be pointed to the paper in which you're describing it.

Your work is very attention-worthy and I certainly hope it receives the recognition it deserves. Should you have enough time to take a look at my essay, your comments are very welcome.

Warm regards,

Alma

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I enjoyed a lot your witty and refreshing essay. I too have mentioned the proof of Fermat's Last, but your most amusing and insightful conclusion about a companion to Wigner's paper didn't even cross my mind.

I think your treatment of time as fundamental is exciting and aiming to fill a gap in the foundations of today's physics. I know you have a book on the topic and I am planning to read it, however I am reasonably sure that the book, as any other topic-focused reading, does not answer a curiosity of mine regarding your view. I would like to know where is your intuition about time stemming from. I mean, when was that moment that your intuition crystallized and what caused it to happen? If you did write about it, I'd be grateful to be pointed to the paper in which you're describing it.

Your work is very attention-worthy and I certainly hope it receives the recognition it deserves. Should you have enough time to take a look at my essay, your comments are very welcome.

Warm regards,

Alma

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Dear Alma,

Thanks for the kind words. It took a long time to arrive at these ideas, which basically come from two different directions. One was teaching the Theory of Relativity year after year, learning to think of it in purely geometrical terms rather than in terms of coordinate systems (Lorentz transformations, etc.). Eventually, one gets intuitions about how the geometrical structure works. The second strand came, while teaching on of these classes, from the realization that the standard way to approach topology is very hard to get a clear conceptual grip on. (I have had many, many students say that this is their reaction when they first learn the theory). I realized when I was trying to teach the standard approach that it does not at all correspond to how I would think of things, and asked myself whether a more visualizable theory could be created. This is the outcome of years of work in that new direction.

The curious thing is that several mathematicians have remarked how natural this way of setting things up is, and can't believe it hasn't been tried before. But so far, no one has pointed out anything similar. It is such a simple idea, in the end, that I am puzzled as well. Maybe it will turn out that someone thought of the basic approach long ago.

Once you put together a completely geometrical understanding of Relativity with this new way to think about geometry, the role on time in Relativity just jumps out at you. That was not something I was aiming at. But this mathematical language fits Relativistic space-time geometry like a glove. It is hard not to think that there is something significant in that. And time takes center stage as the basic ordering principle.

Cheers,

Tim

Thanks for the kind words. It took a long time to arrive at these ideas, which basically come from two different directions. One was teaching the Theory of Relativity year after year, learning to think of it in purely geometrical terms rather than in terms of coordinate systems (Lorentz transformations, etc.). Eventually, one gets intuitions about how the geometrical structure works. The second strand came, while teaching on of these classes, from the realization that the standard way to approach topology is very hard to get a clear conceptual grip on. (I have had many, many students say that this is their reaction when they first learn the theory). I realized when I was trying to teach the standard approach that it does not at all correspond to how I would think of things, and asked myself whether a more visualizable theory could be created. This is the outcome of years of work in that new direction.

The curious thing is that several mathematicians have remarked how natural this way of setting things up is, and can't believe it hasn't been tried before. But so far, no one has pointed out anything similar. It is such a simple idea, in the end, that I am puzzled as well. Maybe it will turn out that someone thought of the basic approach long ago.

Once you put together a completely geometrical understanding of Relativity with this new way to think about geometry, the role on time in Relativity just jumps out at you. That was not something I was aiming at. But this mathematical language fits Relativistic space-time geometry like a glove. It is hard not to think that there is something significant in that. And time takes center stage as the basic ordering principle.

Cheers,

Tim

Dear Tim,

Thank you for your answer as it's a very interesting insight regarding how new theories come to be. I don't find it particularly odd that no one thought of your idea before, because most people don't feel the need to innovate as long as there exists a functional theory, all the more since the effort of creating a logically sound new approach is considerable. Also I'm sure your unique intuition has to do with this, because I think I understand that your approach allows for discreetness, not necessarily requiring infinitesimals.

Cheers,

Alma

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Thank you for your answer as it's a very interesting insight regarding how new theories come to be. I don't find it particularly odd that no one thought of your idea before, because most people don't feel the need to innovate as long as there exists a functional theory, all the more since the effort of creating a logically sound new approach is considerable. Also I'm sure your unique intuition has to do with this, because I think I understand that your approach allows for discreetness, not necessarily requiring infinitesimals.

Cheers,

Alma

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Tim,

I only learned about you existence from watching the “time” video where you indulged the doctrine of Julian Barbour.

I knew ahead of time (no pun intended) that your philosophy would be self-consistent (no surprise there).

So this is just my “plug” to aid in judging your essay. We need people capable of composing the whole picture in an internally logical way, and Tim is one of those people. (Don’t get any ideas that I agree with everything you say.). Not that you care.

En

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I only learned about you existence from watching the “time” video where you indulged the doctrine of Julian Barbour.

I knew ahead of time (no pun intended) that your philosophy would be self-consistent (no surprise there).

So this is just my “plug” to aid in judging your essay. We need people capable of composing the whole picture in an internally logical way, and Tim is one of those people. (Don’t get any ideas that I agree with everything you say.). Not that you care.

En

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Dear En,

I'm glad you liked the video...it was fun to do. And I never imagine anyone agrees with everything I say!

Cheers,

Tim

I'm glad you liked the video...it was fun to do. And I never imagine anyone agrees with everything I say!

Cheers,

Tim

1. Dear Tim,

Your essay was for me the occasion to discover your theory of linear structures, and it was an enchantment. I think it won't be long till I buy your book.

Your text is particularly clear, too.

If I really like your style and your contributed linear structures, I have not really found an answer as to the effectiveness of mathematics, or, as you aptly...

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Your essay was for me the occasion to discover your theory of linear structures, and it was an enchantment. I think it won't be long till I buy your book.

Your text is particularly clear, too.

If I really like your style and your contributed linear structures, I have not really found an answer as to the effectiveness of mathematics, or, as you aptly...

view entire post

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Dear Vincent,

Thanks for the extensive comments.

You seem to object to the idea of some immediate, certain access to the structure of the physical world via perception. Of course that is correct. I'm sure I have not written anything that suggests otherwise. Physical theories are always conjectural.

But it is also clear that some mathematical structures (say Euclidean geometry) were suggested fairly directly by interaction with the world and perception, while others (octonians) were arrived at by abstract considerations fairly far removed from perception. This observation is compatible with the fallibility of perception, perceptual error, etc.

Yes, by "physical space" or "physical space-time" I mean what we wave our arms and legs around in! Physical theories are proposals for the structure of that, and its contents. The aim of physical theory is to give an accurate account of physical space (or space-time) and its contents. What we can alter are the theories, not the object that the theories are designed to describe. And of course no one has "the final word" about whether a physical theory is an accurate description of the physical world. That is the point of being fallibilistic about all theories.

Cheers,

Tim

Thanks for the extensive comments.

You seem to object to the idea of some immediate, certain access to the structure of the physical world via perception. Of course that is correct. I'm sure I have not written anything that suggests otherwise. Physical theories are always conjectural.

But it is also clear that some mathematical structures (say Euclidean geometry) were suggested fairly directly by interaction with the world and perception, while others (octonians) were arrived at by abstract considerations fairly far removed from perception. This observation is compatible with the fallibility of perception, perceptual error, etc.

Yes, by "physical space" or "physical space-time" I mean what we wave our arms and legs around in! Physical theories are proposals for the structure of that, and its contents. The aim of physical theory is to give an accurate account of physical space (or space-time) and its contents. What we can alter are the theories, not the object that the theories are designed to describe. And of course no one has "the final word" about whether a physical theory is an accurate description of the physical world. That is the point of being fallibilistic about all theories.

Cheers,

Tim

Tim Maudlin Apr. 4, 2015: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."

You abandoned this discussion after realizing that no such experiment has ever been done (no lifting and then returning to the floor). Gravitational time dilation has always been measured by measuring the gravitational redshift but the redshift actually confirms the variable speed of light predicted by Newton's emission theory of light:

Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

Pentcho Valev

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You abandoned this discussion after realizing that no such experiment has ever been done (no lifting and then returning to the floor). Gravitational time dilation has always been measured by measuring the gravitational redshift but the redshift actually confirms the variable speed of light predicted by Newton's emission theory of light:

Albert Einstein Institute: "One of the three classical tests for general relativity is the gravitational redshift of light or other forms of electromagnetic radiation. However, in contrast to the other two tests - the gravitational deflection of light and the relativistic perihelion shift -, you do not need general relativity to derive the correct prediction for the gravitational redshift. A combination of Newtonian gravity, a particle theory of light, and the weak equivalence principle (gravitating mass equals inertial mass) suffices. (...) The gravitational redshift was first measured on earth in 1960-65 by Pound, Rebka, and Snider at Harvard University..."

Pentcho Valev

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I stopped the discussion because you do not know the situation with respect to either the predictions or tests of General Relativity. The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different. Your insistence that they are not just demonstrates your mack of understanding of the situation.

"The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different."

Such tests do not and cannot exist. As I have already said, these clocks are atomic oscillators that do not keep record of the time elapsed. So there is no point in lifting one of them and then bringing them together - there is nothing to compare. One can only measure the frequency difference (redshift) and ponder whether it is due to gravitational time dilation or variation of the speed of light with the gravitational potential.

Of course nobody is going to check which of us is telling the truth - you are right by definition. So... good luck!

Pentcho Valev

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Such tests do not and cannot exist. As I have already said, these clocks are atomic oscillators that do not keep record of the time elapsed. So there is no point in lifting one of them and then bringing them together - there is nothing to compare. One can only measure the frequency difference (redshift) and ponder whether it is due to gravitational time dilation or variation of the speed of light with the gravitational potential.

Of course nobody is going to check which of us is telling the truth - you are right by definition. So... good luck!

Pentcho Valev

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1971

http://en.wikipedia.org/wiki/Hafele–Keating_experiment

Such

tests have exited for 44 years.

That is a fact.

If you think they "do not an cannot exists" then you have been proven wrong.

http://en.wikipedia.org/wiki/Hafele–Keating_experiment

Such

tests have exited for 44 years.

That is a fact.

If you think they "do not an cannot exists" then you have been proven wrong.

Dear Tim Maudlin,

I will comment mainly on the first part of your essay. (The second part of the essay, in which you outline your new foundations for geometry, is interesting in it's own right, but it does not seem to address the original question as directly.)

One aspect that I liked is that you distinguish the surprise due to discovering a connection between seemingly distinct parts of mathematics from the wide applicability of mathematics to empirical sciences.

In the first part, you also think about how the world needs to be in order for the counting numbers to be relevant. You give mountains and cells as examples of concepts that are not (always) sharp enough (conceptually) for counting, and atoms as an example of a concept that is. In my view, however, the sharpness of our concepts is a matter of degree. The concept of an 'atom' is still a vague one, to some degree; hence the question of whether something is an atom or not -and whether to count something as an atom or not- may still be ambiguous in certain situations. For instance, one can imagine an electron and a proton: they will always 'feel' each other's electrical field. When, exactly, do they form a hydrogen atom? This question is also relevant in foundations of chemistry (with competing partition schemes in computational chemistry to identify 'atoms in molecules').

Viewed in this way, there are many degrees of freedom in applying mathematics to empirical findings - something which is rarely discussed in relation to the perceived effectiveness of mathematics.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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I will comment mainly on the first part of your essay. (The second part of the essay, in which you outline your new foundations for geometry, is interesting in it's own right, but it does not seem to address the original question as directly.)

One aspect that I liked is that you distinguish the surprise due to discovering a connection between seemingly distinct parts of mathematics from the wide applicability of mathematics to empirical sciences.

In the first part, you also think about how the world needs to be in order for the counting numbers to be relevant. You give mountains and cells as examples of concepts that are not (always) sharp enough (conceptually) for counting, and atoms as an example of a concept that is. In my view, however, the sharpness of our concepts is a matter of degree. The concept of an 'atom' is still a vague one, to some degree; hence the question of whether something is an atom or not -and whether to count something as an atom or not- may still be ambiguous in certain situations. For instance, one can imagine an electron and a proton: they will always 'feel' each other's electrical field. When, exactly, do they form a hydrogen atom? This question is also relevant in foundations of chemistry (with competing partition schemes in computational chemistry to identify 'atoms in molecules').

Viewed in this way, there are many degrees of freedom in applying mathematics to empirical findings - something which is rarely discussed in relation to the perceived effectiveness of mathematics.

Best wishes,

Sylvia Wenmackers - Essay Children of the Cosmos

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Dear Sylvia Wenmackers,

Thanks for the comment. I do completely agree with what you say, which is why also the term "atom" is not used in the formation of fundamental physical law. First it was replaced by reference to electrons, protons and neutrons and then by reference to electrons and quarks. Since atoms are bound states of these things, the concept is only as sharp as "bound state", which is somewhat vague. At this point, "electron" and "quark" appear to be fundamental, and admit of not further analysis. But we may be wrong about that.

Cheers,

Tim

Thanks for the comment. I do completely agree with what you say, which is why also the term "atom" is not used in the formation of fundamental physical law. First it was replaced by reference to electrons, protons and neutrons and then by reference to electrons and quarks. Since atoms are bound states of these things, the concept is only as sharp as "bound state", which is somewhat vague. At this point, "electron" and "quark" appear to be fundamental, and admit of not further analysis. But we may be wrong about that.

Cheers,

Tim

Dear Tim, I very much enjoyed your essay and hope you can clarify one point for me. It seems that your argument suggests that there may be equivalent frameworks for describing physical systems (such as topology or linear structures) but that only one corresponds to physical reality. This suggests that we could be using the wrong mathematics to describe reality purely because distant branches of mathematics that seem unrelated are actually deeply related. How would we ever know then that we had the "right theory"?

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Dear Sara,

I don't think we will ever we absolutely sure we have the right theory. All we can do is formulate different theories, using different mathematical resources, see what sorts of testable, observable behavior they predict, and try to gather data to test them. But there will always be different theories that agree with all the data we have. Choice among these different theories that are consistent with observations will either be made on grounds of simplicity, elegance, etc. or else not be made at all: we will just have to admit that we don't know for sure which, if any, if these theories is correct. But whatever we do, we have to acknowledge that we cannot be certain that any particular theory is right. That's just the situation we are in.

Cheers,

Tim

I don't think we will ever we absolutely sure we have the right theory. All we can do is formulate different theories, using different mathematical resources, see what sorts of testable, observable behavior they predict, and try to gather data to test them. But there will always be different theories that agree with all the data we have. Choice among these different theories that are consistent with observations will either be made on grounds of simplicity, elegance, etc. or else not be made at all: we will just have to admit that we don't know for sure which, if any, if these theories is correct. But whatever we do, we have to acknowledge that we cannot be certain that any particular theory is right. That's just the situation we are in.

Cheers,

Tim

Dear Tim :

Your essay is very interesting,

In my opinion, mathematics that reflects reality, remains a priority for the current science.The Bi-iterative calculation could be the right one, because it has all the characteristics of a pure mathematics.

“NUMBERS”

first are integers

second are geometric shapes

third are physical entities, with...

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Your essay is very interesting,

In my opinion, mathematics that reflects reality, remains a priority for the current science.The Bi-iterative calculation could be the right one, because it has all the characteristics of a pure mathematics.

“NUMBERS”

first are integers

second are geometric shapes

third are physical entities, with...

view entire post

attachments: 1_1_Theorem_1.jpg, 1_1_Theorem_345.jpg

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Tim,

A very interesting essay. I was skeptical at first but your explanation gradually resolved most of my reservations. Then I found just as it was getting to the climax with 'discrete Relativistic space-times' the last chapter was missing!

If you've read any of my recent essays (all finalists) you'll see various aspects of such discrete 'space-time' systems or 'fields' explored. I...

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A very interesting essay. I was skeptical at first but your explanation gradually resolved most of my reservations. Then I found just as it was getting to the climax with 'discrete Relativistic space-times' the last chapter was missing!

If you've read any of my recent essays (all finalists) you'll see various aspects of such discrete 'space-time' systems or 'fields' explored. I...

view entire post

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Dear Peter,

As I'm sure you know, a complete discussion of the presently available evidence in favor of GR would be quite extensive, and go far beyond the three "classic tests", including the rotational rate of binaries, all sorts of precise clock and timing experiments, as everyone always mentions corrections needed for GPS etc. GR should at least be recovered as a limit of a more...

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As I'm sure you know, a complete discussion of the presently available evidence in favor of GR would be quite extensive, and go far beyond the three "classic tests", including the rotational rate of binaries, all sorts of precise clock and timing experiments, as everyone always mentions corrections needed for GPS etc. GR should at least be recovered as a limit of a more...

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"the discussion with Valev was not productive, and irrelevant to my paper in any case. The fact that he continues it despite being asked to stop (see below) is already indicative of a certain state of mind."

Thanks. Kind of you. I did stop the discussion but just found references (see below) disproving the following text of yours:

Tim Maudlin replied on Apr. 4, 2015 @ 23:49 GMT: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."

Pentcho Valev

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Thanks. Kind of you. I did stop the discussion but just found references (see below) disproving the following text of yours:

Tim Maudlin replied on Apr. 4, 2015 @ 23:49 GMT: "Put two high-precision atomic clocks on the floor together Synchronize. Lift one up on a table. Wait a while. Return to the floor and compare synchronization. This has been done. The clocks go out of syntonization, and the amount out is a function of how long the one is up on the table. No redshift or light involved. Experiments at this precision have only been possible recently."

Pentcho Valev

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Tim,

I understand your views, all conventional. You seem to take agreement with convention as always a priori falsification, so are happy to ignore anomalous findings and assume they'll disappear. I do understand as an educator that's the easiest position to maintain.

I must say my approach differs, seeking out and trying to remove or rationalise anomalies and paradoxes. It proved a...

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I understand your views, all conventional. You seem to take agreement with convention as always a priori falsification, so are happy to ignore anomalous findings and assume they'll disappear. I do understand as an educator that's the easiest position to maintain.

I must say my approach differs, seeking out and trying to remove or rationalise anomalies and paradoxes. It proved a...

view entire post

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Tim Maudlin wrote on Apr. 17, 2015 @ 18:08 GMT: "I stopped the discussion because you do not know the situation with respect to either the predictions or tests of General Relativity. The basic scheme of actually comparing clocks has been going on for over three decades. These are not gravitational redshifts. Everyone knows the gravitational redshift is a weak test, but the tests moving and comparing clock readings are just different. Your insistence that they are not just demonstrates your mack of understanding of the situation."

I do understand the situation - you don't:

"A new paper co-authored by U.S. Energy Secretary Steven Chu measures the gravitational redshift, illustrated by the gravity-induced slowing of a clock and sometimes referred to as gravitational time dilation (though users of that term often conflate two separate phenomena), a measurement that jibes with Einstein and that is 10,000 times more precise than its predecessor."

"Einstein's relativity theory states a clock must tick faster at the top of a mountain than at its foot, due to the effects of gravity. “Our performance means that we can measure the gravitational shift when you raise the clock just two centimetres (0.78 inches) on the Earth's surface,” said study co-author Jun Ye."

Pentcho Valev

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I do understand the situation - you don't:

"A new paper co-authored by U.S. Energy Secretary Steven Chu measures the gravitational redshift, illustrated by the gravity-induced slowing of a clock and sometimes referred to as gravitational time dilation (though users of that term often conflate two separate phenomena), a measurement that jibes with Einstein and that is 10,000 times more precise than its predecessor."

"Einstein's relativity theory states a clock must tick faster at the top of a mountain than at its foot, due to the effects of gravity. “Our performance means that we can measure the gravitational shift when you raise the clock just two centimetres (0.78 inches) on the Earth's surface,” said study co-author Jun Ye."

Pentcho Valev

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Dear Tim

I never learn topology, so I do not understand everyting. First I need to learn some background. But, it is known, that special relativity (according to Newtonian physics) defines causality. The now thing is also unsimultaneity. How can you connect your explanation with unsimultaneity?

I think that every new explanation from new aspect can tell a lot, so it seems to me that you have a good essay.

In my essay I speculated, that Pythagora theorem is consequence of kinetic energy conservation in ortogonal direction. Do you have opinion about this?

I thing also that Planck's dimensionless nature of space and time tells a lot ... Maybe still your approach fails to complete explanation.

My essay

Best regards

Janko Kokosar

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I never learn topology, so I do not understand everyting. First I need to learn some background. But, it is known, that special relativity (according to Newtonian physics) defines causality. The now thing is also unsimultaneity. How can you connect your explanation with unsimultaneity?

I think that every new explanation from new aspect can tell a lot, so it seems to me that you have a good essay.

In my essay I speculated, that Pythagora theorem is consequence of kinetic energy conservation in ortogonal direction. Do you have opinion about this?

I thing also that Planck's dimensionless nature of space and time tells a lot ... Maybe still your approach fails to complete explanation.

My essay

Best regards

Janko Kokosar

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